Question about a 12v under voltage protection

I am trying to figure out the best way to approach this. I want to set up a device that runs on 12v (obviously step it down for the ESP board power), but want for it to not even try to power up the board unless there is a minimum voltage. It’ll run from a 12v battery under some cases, but then that battery gets plugged in and while the devices still run off the battery, the battery itself is being constantly charged from 110 so it should stay pretty topped off.

My thought was to put on a relay of sorts that only activates when there is 12v, allowing me to send my 3.3/5v to the board. I don’t want any power to go to the board unless there’s at least 12v available. The problem with relays is the clicking when they engage so I thought perhaps doing with MOSFETs but the input trigger voltage tends to be pretty low on those.

I was hoping to find a way, that when the battery is getting too much under 12v to have something that prevents these boards from further draining that battery to zero.

There’s currently no auto-recharge, i.e., solar, that recharges the battery, it’s all manual - but I will likely add solar as the next phase, which I know puts me in this endless loop of “as soon as the battery charges to 12v everything comes on, discharges below 12 and it’s an evil circle” but that’s more of a tomorrow problem than today problem for me (unless I can prepare now easily for that scenario).

Can MOSFETs be used for this purpose instead of relays, or is there another chip/board I should be eying for this application?

When you say that you don’t want “the board” to power up unless there is 12v available, are you talking about the esp board or the 12v device?

Sorry, I wasn’t very clear on that was I? The ESP board. I just want a gateway where the 12v source connects, that determines if the voltage is 12v or more, then it triggers a passthrough where I’ll step down the 12v for the board. I think I can do it with a relay, I just don’t know if I could do it with MOSFET.