Place a Dutch oven or large pot over medium heat.
Add the butter and allow it to melt completely.
Add the diced yellow onion to the melted butter.
Cook for approximately 4 minutes, stirring occasionally, until the onion begins to soften.
Add the minced garlic and cook for 1 minute, stirring frequently.
Stir in the all-purpose flour until it coats the onion and garlic evenly.
Continue stirring for approximately 2 minutes, or until the flour begins to smell lightly toasted.
Gradually whisk in the chicken stock or vegetable stock until the mixture is smooth.
Whisk in the room-temperature half and half.
Season the soup base with kosher salt and freshly ground black pepper.
Add the chopped broccoli florets, chopped cauliflower florets, shredded carrot, and Dijon mustard.
Stir until the vegetables and mustard are evenly incorporated.
Bring the soup to a gentle simmer.
Simmer for approximately 15 minutes, stirring occasionally, until the broccoli and cauliflower are tender.
Remove the pot from the heat.
Add the grated cheddar cheese by the handful, stirring after each addition until fully melted.
Continue stirring until the soup is smooth, creamy, and evenly combined.
Taste and adjust the seasoning with additional kosher salt and freshly ground black pepper, if needed.
Ladle the soup into bowls.
Serve warm.