Combine first 7 ingredients, 1 egg, and 1/2 teaspoon salt in a food processor; pulse to combine. Divide mixture into 4 equal portions; gently shape each portion into a 1/2-inch-thick patty.
Heat a grill or grill pan over medium-high heat; coat with cooking spray. Grill burgers 4 minutes on each side. Remove burgers from grill; grill rolls, cut sides down, 1 minute or until lightly toasted. Remove buns from grill.
Heat oil in a large skillet over medium-high heat. Gently break remaining 4 eggs into skillet; cook, using a rubber spatula to pull the whites toward the center of each yolk to make a 4-inch circle, 3 minutes or until yolks are just set. Remove eggs from skillet; sprinkle with black pepper and remaining 1/4 teaspoon salt.
Spread mayonnaise onto toasted buns; top bun bottoms evenly with tomato slices, avocado slices, patties, and eggs. Top with salsa, sprouts, and bun tops.