Preheat the oven to 375°F (190°C).
In a large skillet over medium heat, cook the ground beef until browned, breaking it apart with a spoon.
Add the diced onion and minced garlic to the skillet and cook until the onion is translucent, about 3-4 minutes.
Season the beef mixture with salt, black pepper, and dried thyme. Stir well and remove from heat.
In a large mixing bowl, combine the sour cream, milk, and melted butter. Mix until smooth.
Lightly grease a 9×13 inch baking dish.
Arrange half of the sliced potatoes in an even layer on the bottom of the baking dish.
Spread half of the cooked ground beef mixture evenly over the potatoes.
Pour half of the sour cream mixture over the beef layer.
Sprinkle half of the shredded cheddar and mozzarella cheeses over the sour cream layer.
Repeat the layering process with the remaining potatoes, ground beef mixture, sour cream mixture, and cheeses.
Cover the baking dish with aluminum foil and bake in the preheated oven for 45 minutes.
Remove the foil and bake for an additional 15 minutes, or until the potatoes are tender and the cheese is golden and bubbly.
Remove from the oven and let the casserole rest for 10 minutes before serving.
Garnish with chopped fresh parsley and serve warm.