Although not as big as a deer, we were on our way home from the midwest rally and what looked like a possum, ran out into the road from the shoulder. At first it looked like a garbage bag, but going 62mph I came upon it fairly quickly.
I was sad by the results as I hit it, however, I was happy that I did not panic. I tried to get onto the shoulder, mindful of the other lane that was full and watching the construction signs on the shoulder. I did not slam on the brakes, although I did apply them. In the end, it was either us or the animal and I had to choose it. By some grace, there was no damage, but the animal was clearly dead.
It's hard not to, but you have to try not to panic. If I had not been thinking, I would have slammed on the brakes, swerved, etc which I've been told would have been deadly. I think the key is everything in moderation...steering, braking, etc. No quick sharp manuvers, even if it means a head on with a deer or other animal.