Depends on how cold. Your pee comes out of your body pretty warm (about 37C), so it comes out hot and steamy in very cold weather. However, when you have cold enough weather to flash freeze your pee, you probably don't want to have any part of your skin exposed to such coldness.

So, the short answer is no, your pee won't freeze almost immediately when peeing here outdoors in winter. It does melt the snow and paint the snow yellow though.