Melt the unsalted butter in a medium saucepan over medium heat.
Sprinkle in the all-purpose flour and whisk constantly for 1 to 2 minutes, until the roux turns a light golden brown.
Slowly pour in the beef broth in a thin stream while whisking continuously, ensuring each addition is fully incorporated before adding more.
Bring the mixture to a gentle simmer over medium heat, whisking occasionally, and let it simmer for 3 to 5 minutes until it thickens to coat the back of a spoon.
Remove the saucepan from the heat, stir in the salt and black pepper, taste, and adjust seasoning if needed.
Serve the brown gravy hot over mashed potatoes, roasted meats, biscuits, or your preferred dish.