Melt butter over medium heat in skillet, then add flour.
Whisk for 3-4 minutes letting roux brown and cook out flour taste.
Add milk, a cup at a time, whisking for a minute in between cups allowing sauce to barely bubble. Then add second cup and whisk until barely bubbling again.
Remove pan from heat.
Stir in 1 ½ cups of the cheese. (Reserve other half cup for topping).
Pour sauce into sprayed 9 x 11 baking dish (or 2 quart).
Top with shredded Brussels sprouts and toss gently to coat with cheese sauce
Add 2 slices bread to food processor and pulse until fine bread crumbs form.
Mix with remaining ½ cup of cheese and sprinkle evenly over top of gratin.
Melt butter in microwave and drizzle over top of bread crumbs evenly.
Bake for 45 minutes.
Recipe by Dishin & Dishes at https://www.dishinanddishes.com/brussels-sprouts-gratin/