After a freeze, the brown growth on top is insulating whatever is still alive underneath. Cutting it off in January exposes the crown to the next cold snap.
Wait until it pushes
Leave the damage until you see new growth at the base, usually March. Then cut back to it. You will be surprised what comes back.
Check the irrigation before spring
Freezes crack backflow preventers and above-ground valves. Find it in February on your terms, not in June when the yard is wilting.