Preheat the oven to 450 degrees. Spread the potatoes across a large baking sheet, lined with a silpat mat or greased with oil. Drizzle the potatoes with a couple tablespoons of oil and toss thoroughly to coat. Sprinkle generously with salt and pepper. I used at least a teaspoon of salt and a 1/2 teaspoon of pepper. Place in the oven and roast for 35-40 minutes.
Once the potatoes have started roasting, trim and halve the Brussels sprouts. Spread them across a large baking sheet and drizzle with about a tablespoon of olive oil. Toss well to coat. Sprinkle with 1/2 teaspoon of salt and 1/2 teaspoon of pepper. Place them in the oven with the potatoes and roast for 30 minutes, or until they are lightly browned and tender.
Cook and crumble the sausage in a skillet over medium high heat. Drain any grease and discard. Grease a large baking dish (I used a 10x12 lasagna pan) and combine the roasted potatoes, brussels sprouts and sausage. Stir gently to combine. Sprinkle with thyme and stir in the cubes of cheese as evenly throughout the dish as possible. Sprinkle with Pecorino Romano or Parmesan.
Reduce the oven temperature to 400 degrees and bake until the cheese has melted, about 15 minutes. Let rest 10 minutes before serving. (Stir if desired, right before serving.) Enjoy!