Preheat oven to 350 degrees F (175 degrees C). Press pie crust into a 9-inch pie dish.
Melt butter in a saucepan over medium heat. Cook and stir cornstarch, nutmeg, garlic salt, and black pepper into melted butter until cornstarch dissolves, 2 to 3 minutes. Pour milk into cornstarch mixture; cook and stir until thickened and bubbling, about 5 minutes.
Stir Swiss cheese into milk mixture; cook and stir until melted and cheese sauce is smooth, about 5 minutes.
Whisk 1 to 2 tablespoons cheese sauce and eggs together in a bowl until eggs are warmed; pour into cheese sauce and whisk until incorporated, 1 to 2 minutes. Stir ham into cheese sauce; pour into pie shell.
Bake in the preheated oven until pie is set in the middle, 30 to 35 minutes. Let cool slightly before serving, 5 to 10 minutes.