Heat oven to 350°F; butter an 8-inch square dish. Whisk eggs, milk, and pepper; add ham and dried roasted pepper.
Bake in dish 20 to 24 minutes until set and center reaches 165°F. Rest 5 minutes; cut into eight rectangles about 2 by 4 inches.
Meanwhile, toast cut mini-bagel faces in a dry skillet 1 to 2 minutes per batch, keeping outer crust soft. Divide Fontina into eight equal portions.
Fold each egg rectangle into a 2-inch square on a bottom bagel. Add one cheese portion; close.
Place all eight in a baking dish; loosely cover with foil. Bake at 350°F for 5 to 7 minutes until cheese melts. Remove foil before serving two per person.