Alright! So I fished an egg out of the ruby red brine, scooped out the yolk, mixed it with mayo, horseradish grainy mustard, seasoned it with salt and pepper and garnished it with a little chopped green onion. And yum! The pickled egg white added a great vinegar taste, although I didn't taste any distinct beet flavor. Maybe the beets are just included to show off their beautiful color.
My only complaint is that the pickling process seemed to make the whites a little rubbery, which was kinda weird to bite in to. But it's a small price to pay for something so tasty (and easy).
See preceding post for recipe.