Lightweight poly gloves. Thinnest available.
Hand muff, integrated muff, or pass through pockets (to inside of jacket for warmth.)
Hand warmers inside the muff.
If it is super cold I wear a wool glo-mitt on my bow hand with a hand warmer inside the flip top portion. But still just a thin poly glove on the release hand.
Hang the bow on a hook. Deer walks in grab bow off hook and draw back for the shot.
In really cold weather gloves or mittens alone don’t keep the hands warm enough. Too much heat loss at the wrist joint where the glove / mitten meet the jacket sleeve. That - plus - you have to take the mitten off. Removing mittens is a lot of movement, you have to have a place to put them, and if you do it too soon (like when the animal slows down) your hands absolutely freeze. And then you have to either sit there with frozen hands or put the mittens back on. Big heavy mittens or gloves are a definite NO GO for me. Been there, done that. Never again.
For super cold weather (the ridiculous zone), I strap boot blankets around my waist and put my hands inside with a hand warmer. Can bury the arms in up to the mid forearm. Never gotten cold with this approach. A lot of bulk and extra stuff to carry in, but it works.
Tried the fingerless gloves before, but they don’t work particularly well in really cold weather - because it is the tips of the fingers that get cold first - and fingerless gloves don’t help in that regard.