Growing up, my mother always used to bake them.
They do turn out a bit drier than if you deep-fried them, but not overmuch.
They cook for about 15-20 minutes at 400°F (~200 °C), or until golden and crispy on the outside.
Alternately, you can pan-fry them, as other answers have suggested.
Or you can split the difference and oven fry them.
If you're unfamiliar, Google can acquaint you with the process.
Here's a good NPR article to get you started: http://www.npr.org/templates/story/story.php?storyId=130673515
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You can pan fry them if you're careful (6-12mm oil).
It helps to coat the balls with rye flour or similar before frying.
Leave them alone to fry until the bottoms are nicely brown, then turn carefully w a metal spoon.
If you have trouble with them breaking during the turning, you can flatten them from perfect ball shape a bit.
However, that's best done as soon as you put them in the oil.
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Just fry them in a pan with a few tablespoons of oil like you would meatballs, you won't get the same all round browning but it'll do.
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We don't fry, pan-fry, or bake our falafels.
Instead, we put them on a non-stick grill (same grill we use to make pancakes, etc.).
I imagine that a non-stick frying pan would do the same.
We don't use any oil at all.
Depending on your non-stick surface, you may need to spray it with PAM or something similar.
We get a fairly nice browning, but we end up with a shape of a round patty instead of a ball.
We've found that refrigerating the dough for an hour prior to cooking helps a lot in making sure that the batter stays together.
I haven't found them to be any dryer than when I am frying them.
The recipe I used was by the Looneyspoons girls, in their new recipe book.
It worked very well.
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Falafel can be baked in the oven for about 20 minutes using few tablespoons of oil around 200°C (392°F).
You should be careful when forming the balls not to make them too large.
It is better to make them a little bit thin to make sure they are well cooked, and help prevent stomach aches.
After baking or frying, you can use a paper towel to soak up the excess oil.
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lilghtly coated in rapeseed oil, or oil with high smoke point, and air fried
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I see in this  video  how the falafel balls are fully submerged in oil. Are there other ways (e.g. baking) to prepare the falafel balls after the balls are formed? Note: I do not ask for recipes.
Is there a way to make falafel without deep frying?
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<middle-eastern-cuisine><chickpeas>
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You can bake or pan fry them, although you won't get the same even browning as with deep fried. You can bake them in the oven or pan fry them in a little oil.
You can bake them in the oven for about 20 minutes with a little oil. You can pan fry with a couple of tablespoons of oil. You can grill them or air fry them.