As reasonable adults might predict, she goes out without mittens, and sometimes puts her hands it the snow, and regularly cries/screams/complains that her hands are cold. Duh.
I carry her a lot when there's snow on the ground, which avoids the issue of snow in her socks/pants/shoes.
It would be nice to let her go sledding -- we have a nice toddler sled and lots of snow here -- but it seems crazy to put a kid on a sled who's not wearing mittens and boots, like it will inevitably result in freezing misery.
Today I told her she could come play outside with me and her big sister while I shoveled, but only if she wore her boots -- the snow is deep. She said no. We left her inside and went out without her, checking on her every minute or two and giving her the chance to let me put on her boots so she could come out, too. She was stubborn and stayed in the whole time, sometimes crying, sometimes not.
We have many mittens options (all refused, so it's not about color/style), only one pair of boots (refused before they were ever tried on so I don't think it's a comfort issue).
Does she just have to wait to go sledding until she's 3? It's too bad. DD1 loved sledding at age 2.