Egg wash alone won't deter deer, but Deer Off can be safely used on vegetables, but not on the parts you intend to eat (they'll taste terrible). For example, you can spray beans when they're in flower--the spray lasts for up to three months, though not on new growth, of course--and you'd probably safely get beans.

However, here's what I'd do. Put a single strand electric fence around the garden and bait it with peanut butter. Spread the PB on small squares of heavy duty aluminum foil and attach them to the fence with clothes pins. The foil will last a week or so. Deer associate an experience with the place where it happened and it will drive deer away. If you have several herds you may have to repeat until all the deer in the neighborhood have heard about the terrible experience some of them have had.

Watch carefully for signs of deer and repeat when necessary.

I am also about to start my veggie garden; I plan to put up an electric fence. Why would it be necessary to bait it with peanut butter? Isn't it enough that when the dear come to try to eat some veggies they will get shocked?

Thanks!