Heat oven to 375°F. Grease a 9×13-inch baking dish with butter.
Brown the ground beef in a skillet over medium-high heat, 5–6 minutes. Drain excess fat.
Stir in onion and garlic. Cook until the onion softens, 3–4 minutes.
Season with Worcestershire, salt, pepper, and paprika. Remove from heat.
Whisk the soup, milk, and sour cream in a bowl until smooth and creamy.
Arrange half the potato slices in the baking dish. Sprinkle lightly with salt and pepper.
Spread half of the beef mixture over the potatoes.
Pour half of the soup mixture evenly over the beef. Top with about 1 cup cheese.
Repeat the layers with remaining potatoes and beef, then pour on the remaining sauce.
Cover tightly with foil and bake for 40 minutes.
Uncover, sprinkle on the remaining cheese, and bake until potatoes are tender and bubbly, 12–18 minutes.
Rest 10 minutes. Garnish with parsley and serve warm.