Preheat the oven to 350 degrees F. Grease a square baking dish with butter (8 x 8 or 9 x 9 inch) and set aside.

In a large sauté pan over medium heat melt the butter. Cook the diced onion until it softens but does not brown, about 2 to 3 minutes. Throw in the minced garlic and cook while stirring for 1 more minute.

Turn the heat down to medium low and whisk in the flour. Continuously stir until the flour darkens but is not burnt, about 1 to 2 minutes.

Whisk in the milk, salt, and pepper until it slightly thickens, about 1 minute.

Turn the heat off and, using a spatula or wooden spoon, stir in the grated cheese until it melts. Fold in the hash browns until evenly coated.

Transfer to the baking dish being careful to push the thick mixture down on top into one even layer.

Bake for 30 to 35 minutes uncovered, and then turn the heat up to broil for 3 to 4 minutes or until the top turns golden brown. Keep a close eye under the broiler so it doesn't burn! Serve warm and enjoy.