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 over medium heat, cook the ground beef until browned, breaking it apart with a spoon. Drain excess fat.
Add the diced onion and minced garlic to the skillet with the ground beef and cook until the onion is translucent, about 5 minutes.
Season the beef mixture with salt, black pepper, and 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 and bubbly.
Gradually whisk in the milk, continuing to whisk until the sauce thickens and comes to a simmer, about 5 minutes.
Remove the sauce from heat and stir in the shredded cheddar and mozzarella cheeses until melted and smooth.
Layer 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, spreading gently to cover.
Add the remaining sliced potatoes in an even layer on top of the sauce.
Pour the remaining cheese sauce over the top potatoes, spreading evenly.
Sprinkle 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 continue baking uncovered for an additional 25 minutes, or until the potatoes are tender and the top is golden brown.
Let the casserole rest for 10 minutes before serving.