Preheat the oven to 350°F. Spray an 8x8-inch baking dish with cooking spray.
In a small bowl, stir together the oat milk and melted butter. set aside.
In a large bowl, combine the shredded Brussels sprouts, cheddar cheese, flour, garlic, salt, thyme, and black pepper. (Using your hands is a good way to ensure the ingredients are well combined.)
Transfer the mixture to the baking dish then pour the milk/butter mixture evenly over the top.
In a small bowl, combine the breadcrumbs, Parmesan cheese, and olive oil. Sprinkle evenly over the top of the Brussels sprout mixture.
Bake, uncovered, for 35 minutes then serve hot.