In a 10- to 12-inch nonstick frying pan over medium-high heat, stir shallots in 1 tablespoon butter. Stir often until golden, about 3 minutes. Add flour; stir until blended. Remove from heat and add broth and milk; whisk to blend well. Return to heat and whisk until mixture is boiling vigorously. Remove from heat and add cheese and nutmeg; stir until cheese is melted. Pour sauce into a bowl. Rinse and dry pan.
Lightly butter one side of each bread slice and lay buttered sides down on a board. Divide half the cheese sauce equally among 4 slices; spread sauce to edges. Lay ham equally over sauce. Spread mustard equally on unbuttered sides of remaining 4 bread slices; invert each onto a slice topped with ham.
Put frying pan on medium heat. Lay two sandwiches in pan; cover and cook until well browned on the bottom. Turn with a spatula and brown other sides, about 4 minutes total. Transfer sandwiches to a 10- by 15-inch pan and keep warm, uncovered, in a 200° oven. Repeat to cook remaining sandwiches. Spoon remaining cheese sauce equally over sandwiches; spread slightly over edges.
Broil sandwiches about 4 inches from heat until sauce is bubbling and lightly browned, 1 to 1 1/2 minutes.
Nutritional analysis per sandwich.