In a large heavy-bottomed pot or Dutch oven, melt the butter over medium heat. Add the diced onion and a pinch of salt. Cook, stirring occasionally, for 5–6 minutes until softened and translucent. Add the minced garlic and cook for another 60 seconds until fragrant.

Sprinkle the flour over the onion and garlic. Stir constantly with a wooden spoon for 2 full minutes to cook out the raw flour taste. The mixture will look thick and paste-like — that's exactly right.

Slowly pour in the chicken broth a little at a time, whisking constantly to prevent lumps. Once all the broth is incorporated and the mixture is smooth, whisk in the whole milk and heavy cream.

Add the dry mustard powder, smoked paprika, cayenne (if using), salt, and black pepper. Stir to combine and bring the soup to a gentle simmer over medium heat.

Add the broccoli florets and grated carrots. Simmer uncovered for 12–15 minutes, stirring occasionally, until the broccoli is very tender and the soup has thickened.

For the best texture, use an immersion blender to blend about half the soup directly in the pot, leaving the rest chunky. Alternatively, transfer half to a blender, blend smooth, and return to the pot. Stir to combine.

Reduce the heat to the lowest setting — or turn off the heat entirely. Add the freshly grated cheddar cheese in 3–4 handfuls, stirring between each addition until fully melted before adding the next. Never add cheese over high heat.

Stir in the fresh lemon juice. Taste and adjust salt, pepper, and cheese as needed. Ladle into bowls and top with extra shredded cheddar, a sprinkle of paprika, and freshly cracked black pepper. Serve immediately with crusty bread.