Melt butter in a saucepan over medium-low heat. Add flour; cook and stir until mixture is golden-brown and smells like cooked pie crust, about 5 minutes. Add garlic; cook for 30 seconds. Whisk in 1 cup of broth.
Stir remaining broth, ketchup, Worcestershire sauce, and mustard into the mixture in the saucepan. Bring to a simmer, whisking, and reduce heat to low. Cook, stirring occasionally, until thickened, about 20 minutes. Season with salt and pepper.