Preheat a large pot over medium heat. Melt butter, then add onion and garlic. Cook, stirring occasionally, until onion has softened, 3-4 minutes.
Add flour, paprika, and dry mustard to the pot and stir to coat. Season with salt and pepper. Add the broth, a little at a time, stirring constantly, until smooth.
Bring the broth to a low boil, then add broccoli, carrot, and split peas. Lower the heat, cover, and simmer for 25-35 minutes, or until split peas are tender.
Add cheese, and stir until melted, about 5 minutes. Taste and adjust seasoning, if necessary.
Garnish with additional cheese, and enjoy immediately!