I feel for you guys who've been subjected to this. I thought I had it tough a month ago when I cycled through a 50+cm deep puddle of 7C/45F water. I held my pannier clear out the water, rolled my trousers up and took off socks and shoes. I had to let my feet dry first (in the wind) before being able to put my shoes/ socks back on again in order to avoid continuing to have wet footwear.
Personally, I would just choose not to cycle across a frozen lake.