Preheat oven to 425° and line a large baking sheet with parchment paper. Prepare an ice bath in a large bowl.
Blanch brussels sprouts: Bring a large pot of salted water to a boil. Add trimmed brussels sprouts and cook until bright green and very tender, about 10 minutes. Add brussels sprouts to ice bath to cool and stop the cooking process. Drain
In a large bowl, toss the blanched and drained brussels sprouts with olive oil, garlic, and thyme. Place on the large baking sheet with parchment paper. Use the end of a small glass or mason jar press down on the brussels sprouts to smash them into a flat patty. Season each smashed brussels sprout with salt and pepper, then sprinkle mozzarella and Parmesan on top.
Bake until the bottoms of the sprouts are crispy and the cheese is melted and golden, about 10 minutes.