Best way to prototype a TDFN package chip (MAX9814 and MAX9820)?
You can solder those chips on with hot air, liquid flux and a bit of solder paste. They have pads that run up the sides so it's possible to inspect the connections (but not the thermal pad). When the thermal pad is properly soldered, the chip should settle down onto the board and center itself. Get some good quality fine flux-infused solder wick to clear any shorts. 

You should use a proper hot air station, but it's also possible to use a temperature controlled (not the $10 kind, the $200 kind) of heat gun (called a 'heat tool' in Europe, apparently). If use the cheap uncontrolled kind you'll probably fry the board.  

You could also call in a favor from whatever assembly house you are planning to use for production and have them populate the chips. Should not cost much if you are a regular customer. 

Alternately, Maxim offers (somewhat expensive) evaluation boards for both of the chips, and Adafruit offers an (inexpensive) board for the mic amplifier, all of which come with the chips already soldered.