In a large pot, melt the butter over medium heat. Add the shallots and garlic and cook, stirring often, until the shallots start to soften, 3 to 5 minutes.
Add the cauliflower and red pepper and cook, stirring, until the red pepper is tender, about 5 minutes.
Add the flour and stir until well blended. Immediately add the stock and mustard, stirring well to combine. Bring to a boil over medium-high heat, stirring constantly. Reduce the heat to medium-low and simmer, uncovered and stirring occasionally, until the cauliflower is tender, about 15 minutes.
Add the cheese, 1/4 cup at a time, stirring until each addition is melted before adding more. Remove the pot from the heat and set aside.
In a large skillet over medium-high heat, cook the beef, seasoning it with salt and pepper to taste and stirring, until no longer pink, about 8 minutes. You can break up the larger lumps with the spoon or leave it chunkier. Drain off and discard the excess fat, then set aside.
Using an immersion blender in the pot, blend the soup until smooth. Stir in the cooked beef. Reheat the soup gently over medium heat.
Ladle the soup into warm bowls and garnish with your favorite cheeseburger toppings (see sidebar).
Prefer a chunkier soup? Skip the puréeing step.