Arrange the bread on a baking sheet and spread each slice with 1/2 teaspoon mayonnaise.
Heat a large pan over medium-high heat. Place half (12) of the bread slices, mayonnaise-side down, in the pan and toast until browned, about 3 minutes. Return the toasted bread to the baking sheet, toasted side down, and spread 1 teaspoon mustard on each of the untoasted sides.
Fold each slice of ham into thirds, as if folding a letter, then cut in half to yield 12 squares. Place the ham squares on top of the mustard.
Preheat the broiler to high.
Melt the butter in a medium saucepan over medium heat. Add the flour and whisk until combined; cook until light brown, 1 to 2 minutes. Whisking constantly, slowly add the milk and cook until the sauce thickens, about 3 minutes. Whisk in the salt and nutmeg. Add 1 cup shredded Gruyere; whisk until melted and the sauce is smooth, about 1 minute. Remove the Mornay sauce from the heat, spoon 1 tablespoon over each square of ham, and then top with 1 teaspoon of the remaining cheese.
Broil for 1 minute to melt the cheese and toast the remaining 12 slices of bread (these are the tops of your sandwiches). Remove from the oven. Top the sandwiches with the toasted bread and serve.