Preheat the oven to 375°F (190°C). Grease a 9x13-inch or similarly sized baking dish.

Thinly slice the potatoes to about 1/4 inch. Dice the onion and mince the garlic. Arrange slices so they’re uniform.

Heat olive oil in a large skillet over medium heat. Add the onion and garlic and sauté until softened and fragrant, about 3 minutes.

Add the ground beef to the skillet. Break it up and cook until no pink remains and the liquid has mostly evaporated. Season with Italian seasoning, paprika, salt, and pepper. Taste and adjust seasoning.

Layer half of the potato slices in the bottom of the greased baking dish in a single even layer. Spread half of the cooked beef over the potatoes. Sprinkle with half the cheddar cheese.

Repeat with the remaining potatoes and beef. Top with the remaining cheddar. Pour the beef broth evenly over the dish. If using, drizzle the milk over the top.

Cover the dish tightly with foil and bake for 35–40 minutes, until potatoes are tender when pierced with a knife.

Remove the foil, sprinkle the Parmesan over the top, and bake uncovered for another 8–10 minutes until the top is golden and bubbling.

Let the casserole rest for about 5 minutes. Garnish with chopped parsley, cut into squares, and serve warm.