Mix together beef, garlic, salt, and pepper in a large bowl. Using a burger press, form into 4 patties.

Grill burger patties to desired doneness; keep warm.

While burgers are cooking, heat olive oil in a large skillet over medium heat. Saute onion and green pepper in olive oil until tender (about 5 minutes). Remove vegetables to a bowl and keep warm.

Melt butter in a skillet. Break eggs into the skillet and cook to desired doneness (over medium or over hard works best with these burgers).

Top each bun with a burger, a cheese slice, a fried egg, and a spoonful of the vegetable mixture.

Serve immediately