Prepare all vegetables before turning on the stove. Chop broccoli including stems into bite-sized pieces, shred or matchstick the carrots, chop the onion, and mince the garlic. Set everything aside.

In a large soup pot, melt the butter over medium heat. Add the chopped onion along with the salt, pepper, dry mustard powder, cayenne, and smoked paprika. Stir to combine and saute for 4 to 5 minutes until the onion is soft and translucent.

Add the minced garlic and cook for one more minute until fragrant.

Sprinkle the flour over the onion mixture a little at a time, stirring after each addition until fully absorbed and a smooth paste forms.

Using a whisk, slowly pour in the chicken broth a little at a time, stirring constantly after each addition to keep the sauce smooth. Once all the broth is in, continue the same process with the whole milk, whisking until fully combined.

Increase heat to medium-high and bring to a low simmer. Add the broccoli and carrots. Once small bubbles form, reduce heat to low and maintain a gentle simmer for 5 to 10 minutes until the broccoli is just barely tender. Do not allow a rapid boil.

Turn off the heat. Stir in 1/2 cup heavy cream. Add the freshly grated cheddar one cup at a time, stirring until fully melted before adding the next cup. If cheese is not melting, return heat to low briefly until melted, then turn off immediately.

Taste and adjust seasoning with salt, pepper, or more cream as desired. Serve hot topped with extra shredded cheddar and chopped green onions.