Scramble the egg together so the white and yolk are completely blended. Add a bit of salt and pepper.
1 egg
Place in a lightly greased preheated skillet over low to medium-low heat. Cook until the egg begins to set. Push sides in towards the center. Flip and cook until fully cooked through to your liking.
Butter two slices of bread and place butter side down.
2 slices bread, 2 tablespoons butter
Top with meat, cooked egg, and cheese.
2 slices bacon, ham or sausage patty, 1 slice cheese
Place the top slice of bread on with the butter side facing up.
Place the sandwich on a preheated panini press and cook until the bread is nicely toasted and the cheese is melted, about 4-6 minutes.
Remove from press and serve hot.
Notes
Try to keep the egg in one piece, rather than multiple broken pieces like traditional scrambled eggs. One piece stays together better in the sandwich while you're eating it.
To make the sandwich cheesier, add a slice at the bottom of the sandwich and another slice at the top.
Sandwich is best eaten while hot or while still warm.
To make more than one breakfast sandwich
Plan on one egg per sandwich. Cook all the eggs at once and then divide the cooked egg into smaller pieces for the number of sandwiches needed.
Assemble the sandwiches and cook 1-2 sandwiches at a time on your panini press.
Serve right away, or to serve all at once, wrap the cooked breakfast sandwiches in foil and place in a warm, 300°F oven while the other sandwiches cook.