In muffins and quick breads , I have found that you can actually substitute apple sauce for oil or melted butter.
This has worked very well for me!
Use the same amount of apple sauce that you would oil/butter, if not a bit extra.
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There are a lot of considerations to make when substituting for butter since it plays several roles that depend on the baked good.
Creaming solid butter with sugar is essential for the texture of a cake, because that's where you make all the little pockets that air will blow up later.
Anything that you can similarly beat might substitute well.
Personally I'm considering experimenting with bananas in recipes like this.
Baked goods that use baking soda and don't require creaming are good candidates for having their butter replaced, especially if they just require melted butter.
This is where I'd be experimenting with yogurt or bananas or whatever else sounds interesting.
The way the fat melts determines how much a cookie spreads as it cooks.
A fat with a higher melting temp would make taller cookies, while using melted butter would make flatter cookies.
Oil would be a good substitute in recipes requiring melted butter, just remember that butter is 10-20% water.
In pie crusts, pastries, and biscuits, you build up layers of dough and butter when you roll and knead them, and this is what creates a flaky dough.
Lots of recipes use part butter and part lard for their different melting points to balance flakiness and tenderness.
Using any fat-free substitute would probably be disastrous but I haven't experimented.
This is the one place where I really wouldn't consider using bananas, because you need fat to separate the layers.
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Lard has less saturated fat than butter, and it's great in pie crusts.
I can't speak to its other baking applications because I stick to butter for cookies and muffins and such, but experimenting with less butter when combined with lard to produce the total fat called for in such recipes might be worthwhile.
Also, About.com has an article on dairy-free baking that you might find useful.
It discusses when to use oil and when to use margarine (and when margarine is called for, there are some decent alternatives to traditional margarine on the market); the article has other great tips for baking, as well.
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I use fat free Greek yogurt.
I found it as a recommendation when I was looking up Greek yogurt and everything I have made with it came out as good or better than with butter.
Use the same amount
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Just use vegetable or canola oil.
You'll likely need to adjust your measurement a bit, you typically would use less oil than the equivalent amount of butter.
You may need to add a bit more water to compensate for the water present in butter (nearly 20% of butter is water).
Also note that the finished good will be textured differently.
Cookies will tend to be flatter because you cannot cream the sugar into the oil as you can with butter.
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A way to reduce the use of butter is to make a spread combining butter and canola oil.
1/2 cup butter (softened)1/2 cup canola oil Blend until combined, store in a covered dish in the fridge.
Use like butter for baking or as a spread.
It has ~ half the saturated fat as butter and negligible amounts of trans fats.
Provides mono-unsaturated fatty acids and omega-3 fatty acids.
A compromise though and not what you asked.
I have found this to be a decent compromise.
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I typically hate naming specific brand names but Smart Balance is a decent butter substitute for baking and is used in a 1:1 ratio.
Unfortunately, there will always be a texture difference because different fats react differently to heat, specifically how fast it melts.
Shortening, for example, melts between 115 and 117, meaning it melts pretty fast compared to butter, which melts between 90 and 95.
The faster it melts, the flatter your cookie is likely to be.
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You can use coconut oil instead of butter or in addition to butter.
It will flavor the cookies or cake with a very mild flavor of coconut....use the organic brand of coconut oil.
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Earth Balance ( see this site ) has excellent non-dairy, no trans fat butter sticks (1-1 substitute for butter).
I've personally baked with them dozens of times and they work wonderfully for everything from pie crusts to cookies and muffins.
cooking.stackexchange.com/users/17216/david
1 banana can substitue for 1 stick of butter.
I recently tried this for a banana bread recipe & it worked great.
Added extra banana flavor & eliminated a lot of fat & calories.
I recommend using a fresh banana (not too ripe) so the texture is still firm.
cooking.stackexchange.com/users/16969/Amanda
What can I substitute for 1 cup of butter in baking recipes (e.g. cookies, muffins, cakes, etc.)?  I'm looking for something that has less saturated fat (and  also doesn't have trans fat).  Update:  Since baking is less forgiving than cooking (i.e. if you don't use the exact amount of each ingredient the recipe might fail), could you please also include the amount you would need to use to replace  1 cup of butter ?
Butter substitute for 1 cup of butter for baking
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<substitutions><baking><butter>
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Butter plays several roles depending on the baked good, but usually you can replace it with canola/vegetable oil or a mixture of oil and butter. Lard has less saturated fat than butter, and is good in pie crusts. In some cases, butter can be replaced with Greek yoghurt, bananasand even apple sauce in muffins and quick breads. It depends on the recipe you are making. You can replace some of the butter with vegetable oil or lard. Bananas and apple sauce can also be good replacements.
You can replace some of the butter with vegetable oil. Lard also has less fat than butter.  It depends on what you are baking as butter has a different role in different things. Try banana or apple sauce.