Electret microphone amplifier with analog MEMS microphone
No. That breakout has an op amp on it but that max9814 has an input with a bias voltage on it for powering the mic. So you don't want to connect the output of an op amp (which looks like it is biased to 1/2 VCC btw) to some current source. The two will just fight each other. And besides, you don't want to amplify things twice.

So either just connect a regular electret to the max9814 or just connect the mic breakout to your uc. In the later case it's already biased to 1/2 VCC as well. In the former case, note that a regular electret mic is actually a very good mic. Not sure that the mems mic is really that much better. Also, note that it is highly unlikely that your microcontroller is going to be fast enough to do anything serious with audio.