"The combination of the egg and the aioli makes this burger special. Grab a few napkins, it can get deliciously messy!
When assembling the burger I recommend you spread the aioli on the bottom bun so the top bun is available for the egg."
Whisk mayonnaise, 1 large garlic clove, chipotle powder, and pinch of salt together in a small bowl; cover and refrigerate.
Place ground beef, 1 egg, seasoned bread crumbs, 2 garlic cloves, coriander, black pepper, and 1/4 teaspoon salt in a bowl; mix with your hands until just combined. Shape beef mixture into 4 patties.
Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Cook patties on preheated grill until browned on one side, about 6 minutes. Flip patties; continue grilling until no longer pink in the center, about 6 minutes more. An instant-read thermometer inserted into the center should read at least 160 degrees F (71 degrees C). Transfer burgers to a plate; top each burger with a slice of Cheddar cheese and cover plate tightly with foil.
Heat a nonstick skillet over medium heat; spray with cooking spray. Crack remaining eggs into skillet and immediately reduce the heat to low; season eggs with salt and pepper. Cook eggs until whites are set, about 2 minutes; add 1 tablespoon water to skillet, cover, and cook 1 to 2 minutes more.
Spread about 1 tablespoon chipotle aioli on bottom of each hamburger bun; top each bun with 1 lettuce leaf, 1 onion slice, 1 tomato slice, and 1 burger. Add one egg to top of each burger; place top of hamburger bun over egg.