In a large heavy-bottomed pot or Dutch oven, melt the butter over medium heat. Add the diced onion and cook, stirring occasionally, for 5–6 minutes until softened and translucent. Add the minced garlic and cook for 1 minute more until fragrant.

Sprinkle the flour over the onion and garlic mixture. Stir constantly for 1–2 minutes to cook out the raw flour taste and form a light roux. The mixture will look paste-like — this is correct.

Slowly pour in the chicken or vegetable broth while whisking constantly to prevent lumps. Once smooth, add the whole milk and heavy cream, continuing to whisk until fully combined.

Add the broccoli florets and grated carrot to the pot. Stir in the dry mustard powder, smoked paprika, and cayenne if using. Bring the soup to a gentle simmer over medium heat.

Simmer uncovered for 15–20 minutes, stirring occasionally, until the broccoli is very tender and easily pierced with a fork.

Use an immersion blender to partially blend the soup directly in the pot — aim for about half the soup to be blended smooth while leaving visible broccoli pieces for texture. Alternatively, transfer half the soup to a blender, blend until smooth, then return to the pot. For a completely smooth soup, blend it all.

Reduce the heat to low. Add the freshly grated cheddar cheese in two or three handfuls, stirring constantly after each addition until fully melted and smooth. Never let the soup boil after adding the cheese.

Stir in the fresh lemon juice. Taste and season generously with kosher salt and black pepper. Ladle into bowls and top with extra shredded cheddar and a pinch of smoked paprika to serve.