How to Make It
Heat 2 tablespoons of the oil in a nonstick skillet over medium-high. Add eggs, 1 at a time, to skillet; use a spatula to keep them separate. Cook until edges are lightly browned, 3 to 4 minutes. Carefully flip, and cook until yolks are just firm to the touch, about 2 minutes. Transfer to a paper towel-lined plate. Cool 5 minutes. Using scissors, trim eggs, leaving 1/2-inch of white around yolks.
Place beef, mustard, Worcestershire, pepper, and 1 teaspoon of the salt in a large bowl; use hands to thoroughly combine. Divide into 4 (6-ounce) balls. Flatten each ball into a 6-inch circle. Place an egg in center of each patty; top each with a slice of blue cheese, leaving a 1-inch rim. Carefully fold edges of meat over to cover filling, and seal; flatten into 3/4-inch thick patties.
Heat remaining 2 tablespoons oil in nonstick skillet over medium-high. Cook burgers to desired degree of doneness, about 3 minutes per side for medium. Transfer to a paper towel-lined plate.
Combine mayonnaise, vinegar, avocado, and remaining 1 teaspoon salt in a bowl, mashing with a fork until only small chunks of avocado remain.
Spread 1 tablespoon avocado mixture on bottom and top halves of buns. Top bottom bun with a patty, bacon, lettuce, and top halves of buns.