Melt butter in a large pot or Dutch oven over medium heat. Sprinkle in 3 tablespoons plus 1 teaspoon flour. Whisk constantly until the mixture is thick and all the flour is incorporated, 3 to 4 minutes.

Slowly pour cold milk into the butter mixture, whisking constantly until combined. Simmer, stirring occasionally, until slightly thickened, about 5 minutes. Stir in paprika, ground chipotle, ancho chile powder, cayenne pepper, and salt.

Reduce heat to low and add sharp Cheddar cheese, Monterey Jack cheese, and Muenster cheese, a handful at a time; stir until thoroughly melted. Season with salt to taste.