Prep the oven and pan: Preheat your oven to 375°F (190°C). Grease a 9x13-inch baking dish with a little olive oil or butter.
Slice the potatoes: Wash and peel if you prefer.
Slice into thin, even rounds (about 1/8 inch). Thinner slices cook faster and stay tender.
Brown the beef: Heat a large skillet over medium-high heat with a splash of olive oil. Add the ground beef, breaking it up as it cooks.
Add aromatics and season: When the beef is mostly browned, stir in the onion and cook until softened, 4–5 minutes.
Add the garlic, 30 seconds. Season with 1/2 teaspoon salt, 1/4 teaspoon pepper, smoked paprika, and thyme. Stir in Worcestershire.
Drain excess fat if needed.
Mix the sauce: In a bowl, whisk together the cream of mushroom soup, milk, sour cream (if using), and cheddar until smooth. Season with the remaining 1/2 teaspoon salt and 1/4 teaspoon pepper.
Layer the casserole: Spread a thin layer of sauce in the bottom of the baking dish. Add a layer of potatoes, slightly overlapping.
Spoon half the beef mixture on top. Pour a third of the sauce over it. Repeat with another layer of potatoes, remaining beef, and another third of sauce.
Finish with a final potato layer and the remaining sauce. Drizzle with melted butter.
Cover and bake: Cover tightly with foil and bake for 45 minutes. This helps the potatoes steam and turn tender.
Uncover and add cheese: Remove the foil, sprinkle an extra handful of cheddar on top if you like, and bake another 15–20 minutes, until the potatoes are fork-tender and the top is golden and bubbly.
Rest and garnish: Let it rest 10 minutes so the layers set.
Sprinkle with parsley or chives before serving.