This may be made in a skillet or a sandwich press, it will also work well on a George Foreman contact grill --- bottled roasted red peppers, drained may be used in place of the hot pepper rings.
In a shallow dish whisk the eggs with Dijon mustard, basil, salt and pepper.
Place 4 slices of bread on a flat work surface, then place 1 slice of cheese on top of each slice of bread.
Top evenly with smoked turkey slices or ham, then remaining cheese slices.
Top with remaining bread slices.
Dip the sandwiches into the egg mixture turning to coat soaking the bread well.
Heat a large non-stick skillet over medium heat, then brush with a thin layer of butter.
Cook the sandwiches in batches turning once for about 3-4 minutes on each side or until browned and the cheese is melted (brushing the pan with more butter between batches).