Melt butter in a heavy saucepan over low heat; add flour, stirring until smooth. Cook 1 minute, stirring constantly.
Gradually add broth and milk; cook over medium heat, stirring constantly, until thickened and bubbly. Stir in pepper. Add pimiento, white wine, Worcestershire sauce, and pepper sauce; stir well. Cook over medium heat, stirring frequently, until mixture is thoroughly heated. Remove from heat; add cheese, and stir until melted. Serve immediately.