Whisk together eggs, milk, half-and-half, maple syrup, vanilla, nutmeg, and 1/2 teaspoon salt in a bowl. Heat griddle or large nonstick skillet over medium heat; butter. Working in batches, lay 1 slice ham on griddle. Quickly dip 1 slice of bread into egg mixture, then place on top of ham. Cook until golden brown, 2 to 3 minutes per side. Serve warm with maple syrup alongside.