Prepare the grill for direct and indirect cooking over high heat (450° to 550°F).
In a small saucepan over medium-high heat, melt the butter. Add the flour and cook until the mixture turns a light beige color, about 2 minutes, stirring constantly. Whisk in the milk, mustard, and Worcestershire sauce and cook until the mixture thickens, about 2 minutes, whisking occasionally. Add the three cheeses and stir until they are melted. Set aside and reheat gently just before serving.
Brush the cooking grates clean. In a large disposable foil pan combine the beer, onions, mustard, and sugar and then add the brats. Place the pan over direct high heat, close the lid, and bring the liquid to a simmer. Simmer until the brats are evenly colored and no longer look raw, about 20 minutes, turning occasionally. If the liquid starts to boil, temporarily move the pan over indirect heat to prevent the bratwurst from splitting open.
Lower the grill’s temperature to medium heat (350° to 450°F). Transfer the brats to a small disposable foil pan and place the pan over indirect heat. Strain the onions through a colander into the pan with the brats. Put the onions back in their original pan and place the pan over direct medium heat. Close the lid and cook until the onions are very tender, about 8 minutes, stirring occasionally.
After the onions have cooked for about 2 minutes, remove the brats from the liquid, discard the liquid, and grill the brats over direct medium heat until browned, about 6 minutes, turning occasionally. During the last minute of grilling time, toast the buns, cut side down, over direct heat.
Place each brat inside a bun and then top with onions. Spoon the sauce over the onions or serve the sauce on the side.