Preheat your oven to 375°F and grease an 8x8-inch baking dish with 1 tablespoon of the softened butter.
Spread the finely chopped onion in an even layer on the bottom of the prepared baking dish, then arrange half of the thinly sliced potatoes over the onions.
Scatter the diced ham and half of the shredded cheddar cheese evenly over the potato layer, then arrange the remaining potato slices on top.
In a medium bowl, vigorously whisk together the whole milk, heavy cream, eggs, garlic powder, salt, and pepper until the mixture is completely smooth and uniform.
Slowly pour the milk and egg mixture evenly over the layered potatoes, gently shaking the dish to help the liquid settle throughout.
Dot the top of the casserole with the remaining 1 tablespoon of butter, then sprinkle with the remaining cheese and the paprika.
Bake uncovered for 55 to 65 minutes, or until the top is golden brown, the cheese is bubbly, and a knife inserted into the center of the potatoes meets no resistance.
Let the bake rest on a wire rack for 10 minutes before serving to allow the liquid to set for easier slicing.
