Melt the butter in a large soup pot over medium heat. Add the onions and cook, stirring, until softened, about 3-4 minutes. Add the flour to the pot and stir together. Cook, stirring constantly, for about 1 minute.
Add the milk and chicken broth to the pot and stir in. Add the shredded carrot, diced potatoes and then the chopped cauliflower and broccoli. Stir together and season with a bit of salt and pepper. Bring the soup to a boil over medium heat, then reduce the heat to medium-low (or a bit lower) and simmer the soup until the vegetables are tender, about 15 minutes, stirring regularly as it cooks.
Using an immersion blender or a potato masher, lightly process or mash the soup a bit to break up the largest pieces of the vegetables.
Add the cheddar cheese to the soup and stir in until melted. Stir in the pinch of powdered mustard.
Taste the soup and add additional salt as needed. If the soup tastes flat at all, it needs more salt!
Serve immediately with freshly ground black pepper and garnish with additional shredded cheese, if desired.
Notes
Note 1: Russet (baking) potatoes are highly recommended, as they break down well as they cook, which helps to thicken the soup. You could use a yellow-fleshed potato in a pinch, though they tend to remain firmer in the soup and may require more cooking time.Note 2: If you would like to use frozen broccoli and cauliflower, simply reduce the simmering time to avoid the vegetables from becoming mushy. If you would like to use fresh together with frozen, start the fresh in the soup first, then add the frozen one later in the cooking process.Note 3: An aged cheddar will bring the nicest cheesy flavour to the soup. Look for a 1 or 2-year aged cheddar. Store leftover soup in the refrigerator for up to 3 days. This soup is not recommended for freezing.Be sure to read the notes above this Recipe Card, for more tips on making this recipe including substitution suggestions and step-by-step photos that you may find helpful.