Boards that have both a slot-1 and socket-370 only intend one such CPU to be installed, so I doubt they would cooperate with what you're trying to do.
Assuming you could convince it to work, there's also an electrical problem. These type of boards probably only have a single VRM which is routed to both sockets. So you'd be limited to using processors that can run at the same voltage. If they are both drawing power at the same time, you'd also have to look at the combined current draw.
The requirement to have matched voltages would prevent you from safely using a combo like you suggested - a 2.8V Klamath alongside a ~1.7v Coppermine. The Deschutes P2s run at 2.0V, which a Coppermine *might* tolerate (I don't know), but I don't think many Deschutes are unlocked (if any). So basically, I think the single voltage limitation would really kill this idea.
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I believe that dual Slot-1 boards generally have a separate voltage regulator for each socket, so they should get along with having different voltages for each CPU. This is definitely something to verify for a specific board though.
These boards typically have a BIOS option to enable only one CPU, but I've never seen one that lets you control which CPU it uses. I'd be surprised if there are any that let you manually make that choice.
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I think the most feasible way to pull this off would be by modifying a dual slot-1 board.
It would need to be a board that will happily boot a single CPU from either CPU socket. You'd probably also want it to tolerate mismatched CPUs in dual mode. Some of them are tolerant of these things, and some are picky.
With such a board, you'd then need to find some way to modify it so that you can disable either of the CPUs from being detected. I'm not sure what the simplest way to do that would be.
One method might be to intercept the VID pins (I think there's 5 of them). Route them through a switch so that you can disconnect these pins from being detected by the associated VRM. When the VRM sees all those signals as floating, it should interpret this as no CPU present and disable power to that CPU. Presumably, then, this would leave the system booting with the other CPU as a single.
There might be a simpler way involving just 1 or 2 pins. Some investigation into Intel datasheets would be needed.
You might not even need to intercept CPU pins. Maybe intercepting something upstream at the voltage regulator IC would also work.
Configure the BIOS to run in single CPU mode. Use the switch on the "primary" CPU that the system normally defaults to. When that CPU is switched on, it will have precedence. When it's switched off, the other CPU will become the one that boots.
This is speculation of course, and I may easily be overlooking something.