You can get PID controllers that will drive a solid state relay (SSR), which can be sized for virtually any load. I bought 25 amp SSRs with heat sinks for around $15 IIRC. Good practice is to allow a safety factor of two, so 12.5A (1500W) would be a reasonable load for these. Bigger heat sinks would allow operation closer to rating. Driving my 3-4 amp dorm fridge the SSR doesn't even get warm.
I've had an issue with thermocouples being very sensitive. It sounds like a good thing, but in practice you want a little hysteresis to keep the controller from cycling too quickly. Undamped, the controller cycles the compressor very rapidly. Put the thermocouple in a test tube of water or use an RTD (which is less sensitive)
To me, a PID is overkill for temperature control. A Johnson controller is accurate enough for our purposes and much easier to set up. I like the PID humidity control, though. There doesn't seem to be a Johnson temperature controller equivalent.
Tiarella, I am not too far from you - about 45 minutes north of Albany, NY. I'd help you set up for a little goat milk, or a just day on the farm.