1. In a mixing bowl, combine mayonnaise, chipotle peppers, lime juice, and garlic. Mix well and set aside.
2. Form ground beef into 4 patties. Season with salt and pepper.
3. Heat a skillet over medium-high heat. Cook patties for about 4-5 minutes per side, or until desired doneness. Add cheese slices on top during the last minute of cooking.
4. In the same skillet, melt butter and cook eggs sunny side up.
5. Toast burger buns in the skillet until golden brown.
6. Assemble the burgers: Spread chipotle aioli on the bottom bun, place the patty with cheese, top with a sunny side up egg, and cover with the top bun.