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 dry mustard and flour to the pot and stir to coat. Season with salt and pepper.
Add the broth, a little at a time, stirring constantly, until smooth. It will be very thick initially, but will thin out as more broth is added.
Bring the broth to a low boil, then add broccoli and split peas. Lower heat, cover, and simmer for 25-35 minutes, or until split peas are tender, stirring occasionally.
If you prefer a smoother soup, use a potato masher or stick/immersion blender to purée the soup. You can also leave the soup as is if you prefer more texture. At this stage, you can serve the soup as is, or you can freeze it. Then thaw, reheat the soup, add the milk and cheese and serve.
For a creamy, cheesy soup, stir in the milk and then the cheese.
Taste and adjust seasoning, if necessary, and enjoy!