When it gets to the low 20's F I switch from the ski gloves to leather goose down mittens. I can (carefully) shift brifters and rapid fire levers. I got my mittens around 1970. First for skiing, then motorcycles, then bicycles. On a bicycle they are too warm over low 30 F. It takes a while, just like any clothing on a bike. You need to start out a little cold, so as you ride, you don't get too hot. Many people I know expect to be warm at the start. That's OK for maybe a mile. In deep snow much less.
If one needs to take off your mittens or gloves, tuck the gauntlet of one glove into the other. That works for a very short time. Maybe just as couple of sips of hot coffee If it's going to be longer put the gloves under your jacket. The fastest way I warm my hands is riding hard.