Vegans often replace egg with ground flaxseed or ground chia seeds mixed with cold water. No salmonella risk, there...although I heard a little bit before about e. coli in vegetables...
Applesauce is also a good substitute binding agent, although the resulting dough might taste a little bit like apple.
Pureed frozen bananas, I've heard, make a great substitute for eggs and cream in homemade ice cream.
I did sometimes wonder how cookie dough ice cream eliminated the risk of salmonella poisoning...