Sounds like a Turban variety. They are early-maturing and early to break dormancy. I planted ~8 varieties between late October and late November, and the Turbans are first up for me too.

I would vote for no cutting. Wait until the ground is frozen or nearly so, and mulch them up. They should already have good root systems, so heave shouldn't be an issue; you just want to protect them from extreme cold (i.e. The tips may burn a little, but I think that is better than cutting the whole top. I suspect a cut plant would be more vulnerable to disease than a frost-burned one.

And don't forget to pull away the mulch in early spring to warm and dry the soil.

No, don't cut back! I would wait on the mulch unless it is a lite mulch with the leaves still exposed to the sun until very low freezing temps are predicted. They will take some pretty low temps. The leaves are feeding the bulbs. Let them wilt before mulching unless the temps are going to be very low and freeze the ground.