I saw a recommendation on 30 Minute Meals to place each one in a muffin tin (they were sized so they stuck out the top), and then bake them.
You'd have to find the right sized tray for your size of meatballs, though.
I'm guessing you'd also want a large tray underneath, though, or you risk dripping grease inside the oven. ...
you could probably also just put 'em on a sheet tray and bake 'em, and you'd only get one flat side, rather than from each time you turned them in the pan.
cooking.stackexchange.com/users/67/Joe
Personally, I wouldn't worry too much about how they look, but rather how they taste!
In some sense there is a trade off between the two.
You can initially mold them into a pretty round shape and then cook them them immediately in the sauce; this way will retain their round shape.
However, the more effective way (and the authentic Italian method) is to bake them in an oven first with a bit of olive oil.
This cooks the meatballs nicely and gives their surface a lovely texture and brown colour.
It is however inevitable that they will lose some of their roundness and flatten at the bottom (depending on their firmness), as you state.
A small price to pay for the taste however, I think!
cooking.stackexchange.com/users/1123/Noldorin
Make them smaller, for one.
For another, does your recipe contain much added liquid?
That makes the raw product softer and more likely to squish.
Third, do you press them while they're in the pan?
Don't do that ;)
Echoing advice above, too; baking will maintain roundness better, as will deep-frying.
cooking.stackexchange.com/users/None/
I'd imagine frequently rolling them around in the pan as you sear them would help.
cooking.stackexchange.com/users/219/ceejayoz
You can bake them in the oven on racks (think cookie cooling racks).
This way they brown evenly all around
and they don't change shape.
The downside is more dishes...
cooking.stackexchange.com/users/229/Trey Jackson
I nuke them in a microwave until the proteins bond, then fry in pan.
This is probably heresy, but I get round, tasty meatballs every time.
cooking.stackexchange.com/users/5455/Wayfaring Stranger
After seeing this  question  I started to think more about my meatballs and wonder how can I keep them round?   I have always been under the impression that you should first brown the outside and then cook them through.  Problem is during the browning in the frying pan I end up with flat sides.  What can I do?  Help my meatballs!
How to keep a meatball round?
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<meatballs>
cooking.stackexchange.com/questions/7975
cooking.stackexchange.com/users/1511/Chris
You can bake them in muffin tins or on a cooling rack with a bit of olive oil, or cook them in the microwave until the proteins bond before frying them. Smaller meatballs are also easier to keep round. You can cook them in the oven or microwave first before frying.
Try baking them in the oven instead. Making them smaller will help. Try cooking them slightly in the microwave.