bbal123 wrote:
In both upstairs and downstairs bathrooms, whenever we flush the
toilets, and then immediately go to the sink to wash our hands, the
water coming out of what is supposed to be the cold faucet is very hot.
After about 30 seconds, it will go back to cold.

SNIP

Could be feedback from a water heater connection.
But can also be caused by faulty Single-Handle faucets
or by a washing machine inlet solenoid valve.

Try and isolate fixtures like these (byclosing shutoffs)
and see if the problem disappears.

If not, then look for more complicated causes.

Jim