Lightly brush all surfaces of the bread with your choice of oil.
Assemble sandwiches, layering 1 1/2 ounces of the ham, approximately 2/3 cup scrambled eggs, and 1 slice cheese on cut side of each bread bottom. Close with roll top.
Cook on well-greased grill or in a Panini press until cheese melts and sandwich is browned and heated through. Serve immediately.
In a large bowl, beat the eggs with a fork until well blended, then add the grated butter, mixing until well combined. The butter will emulsify into the eggs and keep them fluffy and tender, even when cooked through.
Heat a nonstick skillet over medium-low heat. Spray with nonstick spray. Pour the egg and butter mixture into the pan. Using a heat-proof rubber spatula to scramble the egg mixture until firm throughout with no visible egg liquid remaining, and large curds have formed. Season to taste with salt and pepper.
Note: Do not let uncooked egg mixture remain at room temperature for longer than one hour (including preparation and service time).