Preheat oven to 400°F. Grease a 1.5-2 quart baking dish (9-10 inch square pan will work nicely) with butter or cooking spray.
In a large skillet, melt 1 tablespoon of butter over medium-high heat. Add the bread crumbs and stir quickly to coat. Saute with stirring frequently for about 2 minutes, until toasted. Remove from the heat, transfer to a plate, and let cool.
Melt 2 tablespoons of the butter (in the same skillet used for the breadcrumbs) over medium heat. Whisk in the flour until smooth. Cook, stirring, for 1 minute. Add the milk, salt, garlic powder, onion powder, and pepper. Bring to a boil while whisking constantly.
Lower the heat and simmer, stirring frequently until thickened, about 10 minutes. Remove from the heat and stir in the thyme. Taste and adjust salt as needed.
While the sauce is simmering, trim and thinly slice the Brussels sprouts. Place the sprouts in the greased baking dish, sprinkle with half of the cheese, and pour sauce over the cheese. Top with the remaining cheese and sprinkle with the toasted bread crumbs.
Bake until lightly browned and bubbly, about 24-28 minutes. Remove from the oven and stir gently to make sure everything is coated in the sauce. Return to the oven for about 10 minutes. Sprinkle with parsley and serve warm.