Prepare. Preheat the oven to 375°F and prep your baking dish well for nonstick.
Blanch. To blanch the sprouts, bring a large pot of salted water to a boil. Carefully add the brussels sprouts to the pot and cook until slightly tender (about five minutes). Then drain the spouts and transfer them to your prepared baking dish.
Sauté In a large skillet over medium heat, heat the olive oil for one minute. Add the chopped onion to the skillet and sauté until translucent ( about three minutes). Then add the garlic and mushrooms to the skillet and sauté until the mushrooms are cooked (about three minutes).
Sauce. To make the sauce, to the skillet with the sautéed ingredients, stir in the seasonings (thyme, salt, and pepper) and the flour and cook the ingredients for another minute. Then gradually pour in the milk and broth, whisking constantly until the mixture thickens and becomes smooth (about three to four minutes).
Cheese. Once your sauce has thickened, you can stir in the cheese and then pour or use a ladle to top the the brussels sprouts in the dish with the sauce.
Topping. In a medium sized bowl, combine together the Parmesan cheese, breadcrumbs, and melted butter, mixing until the breadcrumbs are evenly coated with the other ingredients. Evenly spread the breadcrumbs over the ingredients in the baking dish.
Bake. Bake the casserole 20-25 minutes, or until the casserole is bubbly and the top is golden brown.