Create the Cheese Sauce: In another medium saucepan, melt butter over medium-high heat. Once the butter is melted and slightly bubbly, add flour. Cook, stirring constantly with a whisk, until the mixture forms a paste-like consistency and turns light golden brown, about 5 minutes. Gradually whisk in the warm milk and bring to a simmer over medium heat. Cook and stir continuously until the sauce thickens and becomes smooth, about 10-15 minutes. Reduce heat to low and stir in the shredded cheese until melted and fully incorporated.