As GBExpat and Jeff White indicated, in the first two replies, we should obtain a winter cover garment with suitable pockets, that can lessen the need for us to wear thick gloves. The pockets can accommodate bare or thinly-gloves hands, and if we opt for it, a small handgun can be accommodated in a suitably-deep pocket. I cannot advise on any specific coat, for actual pocket carry, because manufacturers have tended to change to ever-smaller, ever-flimsier pockets, and it has been a number of years since I bought my last real winter parka.
The quick-and-easy answer for gloves that allow shooting, are Mechanix gloves. Mechanics need to handle their their tools, with dexterity. Finding truly best gloves can be a lengthy endeavor
This last cold blast did not compel me to get the winter coats out of the closet, because, being retired, and having already prepared, I did not need to stay outdoors for very long. My solution was good ol’ pocket carry, of a snub-gun, in a left trousers’ pocket, plus, what some might consider summer carry, a Hill People Gear Belt Pack, worn over all layers, but in a way that kept the hoody’s pockets accessible. I generally wore three three layers, on my upper body, when walking the dogs, and kept one hand at a time in a pocket of the outer hoody. A 2” S&W Model 64 K-Frame revolver was in the Belt Pack, oriented for right-hand draw, where I could reach it without having to first clear away any clothing.
The Hill People Gear Belt Pack has a quite long belt, and could be worn over a true winter coat.
Importantly, my three layers made it less necessary to wear gloves. When it was really cold and windy, the middle layer was a really warm sweater, that did a good job of keeping my neck covered. The hoody kept the wind from trying to blow through the sweater, and into my ears. To stay out longer, I would have first added a beanie-like hat, under the hood. Keeping the head, neck, and torso warm will minimize the need for thick gloves.