Preheat a panini style grill (can also use a non-stick pan with a lid). Lightly grease with non-stick spray.
Heat half of the oil in a non-stick pan over medium heat. Brown the ham slices on both sides. Then set aside.
Heat the remaining oil in the non-stick pan over medium heat. Crack the eggs into the pan. Cook until the white of the egg becomes firm, break the yolk, and gently flip over to cook for 1 minute longer. Season with salt and pepper.
In a bowl whisk the mayo and garlic. Season with salt and pepper to taste. Then spread onto both slices of the bread.
Layer the bread with half slice of cheese, ham, eggs, remaining half slice of cheese, and top with bread slice.
If using a panini style grill, lightly spray with non-stick spray and cook the sandwich until it is golden brown and cheese is melted. (If using a pan on the stovetop, add about a teaspoon of oil or more if needed, then the sandwich and cover with a lid. Cook on both sides until the bread is golden brown and the cheese is melted.)