Baking madeleines without burning them
If the insides aren't cooked when the outsides are, you're cooking too hot. Lower the baking temperature... try a batch at 300F for 30 minutes and test them. If that doesn't work, drop it another 15-20 F and extend the time a bit.

It's all a matter of getting the right balance between heat to cause browning and time to let the inside cook completely.