Step 1Preheat oven to 425° and line a large baking sheet with parchment paper. Prepare an ice bath in a large bowl.
Step 2Blanch Brussels sprouts: Bring a large pot of salted water to a boil. Add Brussels sprouts and cook until bright green and very tender, 8 to 10 minutes. Add Brussels sprouts to ice bath to cool then drain.
Step 3On a large baking sheet, toss blanched Brussels sprouts with oil, garlic, and thyme. Using the end of a small glass or mason jar, press down on 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.
Step 4Bake until bottoms of sprouts are crispy and cheese is melty and golden, 20 to 25 minutes.