Short answer is none (to my knowledge). The Expobar Minore (a dual-boiler machine) has some kind of digital controller on it but I dont think its a PID (others could confirm) and thats closer to $2.5k anyway. Most machines in the price bracket youre talking about are heat exchanger machines, which (broadly) rely on a heap of water at the right temperature and a lot of brass to provide thermostability. While in general they do that very well, the downside is that adjusting the setpoint isnt as easy as pushing a button on a PID. However if its thermostability youre after as opposed to the requirement to regularly change the setpoint, then most HX or a DB machines will be fine.
HX machines are governed by a pressurestat, not a thermostat, and the benefit of a pressurestat is that the pressure inside the boiler is the same everywhere, whereas temperature is not. The big problem with thermostats is that to one degree or another theres a lot of thermal lag between the element creating the heat and that heat being transmitted through the water and the boiler casing up to the thermostat. So by the time the thermostat cuts out theres a lot more heat in the boiler than there should be; the same applies on the downward temperature ramp. Add to this the fact that most thermostats have a 20-30 deg C of hysteresis (deadband, i.e. it turns off at say 110C but doesnt turn back on until 80C) and youve got a system with a constantly fluctuating temperature, which makes extractions rather hit and miss. A PID attempts to get around that by learning how much thermal lag is in the system and controlling the heater power so it gets to about the right temperature once all the heat has made its way to the element - it will turn the heater power off well before it reaches the setpoint, and sometimes before any change in temperature has even been detected! On my PIDd Silvia this reduces the hysteresis to about 0.5 deg C.
So the moral of the story is, a HX machine will most likely give you more thermostability than a single boiler machine, without the need for a PID.