Another N00B question

I have a HERMS system with my coil in my HLT. I currently monitor the temp of the mash and switch the pump on/ off to recirculate the wort as needed to maintain mash temp. My mash tun is an insulated keg and I typically only have to turn it on once maybe twice at the most during a one hour rest. My temps usually return to where they need to be in a few minutes. Occasionally I overshoot because I typically have buddies over on brewday and they break my concentration. I am interested in using a PID to monitor the temp of the mash and turn the pump on and off as needed. It seems like most of the setups I see here and on other sites recirculate continuously and use the PID to turn on/off the element/ burner for the bath the coil is in or the element for a RIMS.

Is my idea feasible? Can someone recommend the parts I need. I was going to attempt it with a johnson controls temperature controller that I have but it seems to have stopped working. I assume I will need a relay of some sort.

I saw something kinda like that In BYO last month I believe, but the coil was in the mash rather than hlt. If your hlt is not temp controlled it seems like it would be hard not to overshoot unless you had a mixer in the mash or something. Best thing for herms is run your pump all the time and pid control your hlt heat. Otherwise, to avoid dealing with heat source control, You could move your coil to your mash tun and pid the pump recirculating hlt water through the coil in the mash and keep a low flame on the hlt.