2001 Chevy Impala temperature gauge

I would start with the sending unit
and it wire

The correct way is to have it hooked to a scanner and read engine data
if it showing you have a good engine temp then you move on to check the cluster gauges
and check the BCM
most I ve seen is the thermostat
but usually that follow with check engine code 128 or so
or the coolant level module stuck.
But also check under hood around the thermostat housing for the temp switch and make sure is OK