Season burgers with salt and pepper, and set aside. Whisk olive oil, balsamic vinegar, and mustard together. Add just a little bit of salt and pepper, and toss with the arugula. Set aside. Cook the burgers to the desired doneness. While burgers are cooking, cook eggs over easy, or sunny side up. To build the burger, spread both sides of the bun with mayonnaise to taste, place a small pile of the arugula on the bottom, and then place burger on top. Place egg on top of the burger and then lettuce, tomatoes, and pickles to desired taste.