Preheat oven to 400°F. Coat a 6-cup jumbo muffin tin with nonstick cooking spray. Toss potatoes, butter, thyme, salt and garlic together in a large bowl until coated.
Place 3 potato slices in the bottom of each muffin cup. Top with 1 slice of ham, a few tablespoons of Gouda-Parmesan cheese, and 3 additional potato slices. Repeat layers 1 more time, ending with a layer of potato slices.
Bake for 30 minutes, until potatoes are tender and golden-brown. Top with remaining Gouda-Parmesan cheese. Bake until cheese is melted and golden-brown, about 5-10 minutes. Gently run a small knife around edge of each potato stack to help loosen from tin. Garnish with thyme, and serve.