Step 1: Preheat the oven to 375 degrees F.
Step 2: In a medium-sized mixing bowl, whisk the half and half with the pumpkin, spices and salt.
Step 3: Peel the potatoes and slice them as thinly as possible. You can use a food processor with the slicing attachment, or a mandoline. (The thicker the slices, the longer the cooking time will be.)
Step 4: Add the potato slices to the pumpkin mixture and use your hands to make sure all of the slices are separated and well coated.
Step 5: Use about 1 tablespoon of the melted butter to coat the bottom and sides of an approximately 8 x 8 x 3 inch baking pan.
Step 6: Add one layer of potatoes to the pan and sprinkle with a bit of salt and pepper, followed by about 1/4 cup of the cheese. Repeat this process twice more, finishing with a final layer of 1/4 cup of cheese.
Step 7: If there is any excess pumpkin mixture in the bowl, pour it over the potatoes.
Step 8: Drizzle the remaining 3 tablespoons of melted butter over the top evenly.
Step 9: Place the pan in the preheated 375 degree F oven, and bake until the top is golden brown and the potatoes are tender, 30 to 45 minutes. (Cooking time depends on the thickness of the potato slices.)
Step 10: Let the gratin rest for at least 45 minutes before serving.
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