In a bowl whisk the eggs, milk, salt, and black pepper. Melt the 1/2 tablespoon butter in a small skillet. Add the eggs and cook, stirring to scramble, about 3 minutes for medium-cooked. Set aside.
Prepare the grill for direct cooking over medium heat (350° to 450°F). Preheat the griddle over direct medium heat for 5 minutes.
Lightly butter one side of each bread slice. Place the bread, butter-side down, on a plate. Lightly spread the top sides of the bread slices with mustard. Arrange the cheese on two pieces of bread. Layer with the ham, eggs, and avocado.
Place the 2 remaining pieces of bread, butter-side down, on the griddle. Toast until golden, 1 to 2 minutes. Remove from the griddle. Place the layered sandwich bottoms on the griddle. Top each sandwich with a toasted bread slice. Grill, with the lid closed, until the cheese melts and the bottoms are golden brown, 2 to 3 minutes. Using a spatula, transfer the sandwiches to serving plates. Serve at once.
