Preheat the oven to 375°F (190°C). Grease a 9×13-inch baking dish with butter or non-stick spray.
In a large skillet, heat olive oil over medium heat. Add the chopped onion and cook until translucent, about 4 minutes.
Add the minced garlic and cook for another 1 minute until fragrant.
Add the ground beef to the skillet. Cook, breaking it up with a spoon, until browned and cooked through, about 7-8 minutes. Drain excess fat if necessary.
Season the beef mixture with 1/2 teaspoon salt, 1/4 teaspoon black pepper, and 1/2 teaspoon dried thyme. Stir well and remove from heat.
In a medium saucepan, melt the butter over medium heat. Whisk in the flour and cook for 1-2 minutes until lightly golden, stirring constantly.
Gradually whisk in the milk and beef broth. Continue whisking until the sauce thickens and starts to bubble, about 5-7 minutes.
Remove the sauce from heat and stir in 1 1/2 cups shredded cheddar cheese, 1/4 teaspoon black pepper, 1 teaspoon salt, and paprika until melted and smooth.
Arrange half of the sliced potatoes evenly in the prepared baking dish.
Spread the cooked ground beef mixture evenly over the potatoes.
Pour half of the cheese sauce over the beef layer.
Add the remaining sliced potatoes on top, spreading them evenly.
Pour the remaining cheese sauce over the top layer of potatoes.
Sprinkle the remaining 1/2 cup shredded cheddar cheese and the grated Parmesan cheese evenly over the casserole.
Cover the baking dish tightly with aluminum foil and bake for 45 minutes.
Remove the foil and bake uncovered for an additional 20-25 minutes, or until the potatoes are tender and the top is golden brown and bubbly.
Remove from the oven and let the casserole rest for 10 minutes before serving.
Garnish with chopped fresh parsley and serve warm.