Slice each roll horizontally but not all the way through if you prefer a hinge. Spread about 1/2 teaspoon of softened butter on each cut face. Heat a skillet or griddle over medium heat. Place rolls cut-side down and toast 2–3 minutes until golden and slightly crisp. Remove and keep warm.

Egg filling

Mix eggs

Crack the eggs into a bowl, add the milk, salt, and pepper. Whisk vigorously until homogeneous and slightly frothy (about 30 seconds).

Cook eggs

Warm a nonstick skillet over medium-low heat and add 1 tablespoon unsalted butter. Once the butter melts and foams but is not browned, pour in the egg mixture. Let eggs sit undisturbed for 10–15 seconds, then using a silicone spatula gently push cooked edges toward the center, tilting the pan to let uncooked egg flow underneath. Continue folding and gently stirring until eggs are mostly set but still soft and slightly custardy, about 2–3 minutes total. Remove from heat (eggs will finish setting off the heat).

Portion eggs

Divide the scrambled eggs into 4 equal portions to assemble on the rolls.

Cheese & toppings

Assemble sandwiches

Place one cheddar slice on the bottom half of each toasted roll. Top each slice of cheese with one cooked sausage patty (or two bacon slices) and one portion of the soft scrambled eggs. If using spinach, place a small handful on top of the eggs. Add hot sauce or ketchup to taste.

Melt cheese and finish

Return assembled bottoms (with cheese) to the warm skillet over low heat for about 20–30 seconds, cover briefly with a lid or a metal bowl to help the cheese melt. When cheese is melted, place the top roll on each sandwich and press lightly.

Serve

Cut each sandwich in half if desired and serve immediately while warm.