I currently have this setup and would like to add automatic control over the PWM based on a thermistor reading. Any insights?
Your info is basically correct.
Unlike motor, Peltier needs no flyback diode as it does not generate harmful back EMF.
Low side switching means connection is Batt_pos, load, D of MOSFET and S of MOSFET to Batt_Neg. Send signal High to G for on. Low for off.
To build from discrete MOSFET, I do not have full circuit on hands, but, you should get tons of tutorial from web.
Full H bridge in this diagram is for hot/cold control. For hot-only, your example is fine.
Given thermal mass of typical system, simple on/off control already give you sub-degree centigrade accuracy. Since PWM comes with no extra cost in your example heat-only setup, one try either.
At 7A, better use direct and thicker wiring for the Peltier, other than the example which use breadboard to carry bulk current.
Since the post asks for open idea, how about using H bridge for heat-cold control for even better sub-degree accuracy, either if you really need that accuracy or try it as a learning experiment.
Your device needs 7A, high current H bridge is relatively expensive while single MOSFET heat-only control is less than one US dollar.