Measure out all ingredients and prepare your vegetables: dice the onions into 1/4-inch pieces, cut broccoli florets into 1-inch pieces, and shred the carrots. In a large pot over medium heat, melt the butter completely. Once melted and foaming slightly, add the diced onions and sauté for 2-3 minutes until they become translucent and fragrant—this develops the soup's flavor foundation. I like to let the onions soften fully before adding flour so they release their sweetness into the broth.

Sprinkle the flour over the softened onions and stir constantly for 1-2 minutes to create a roux (the flour will coat the onions and begin to cook). This step is crucial because it prevents lumps from forming and helps thicken the soup later. Add the broccoli and carrots to the roux, stirring well so the vegetables get coated. Cook for 3-4 minutes, stirring occasionally, to start softening the broccoli and allow the vegetables to develop slightly caramelized edges.

Pour in the chicken broth, milk, and heavy cream all at once, stirring vigorously to break up any flour lumps and create a smooth base. Add the kosher salt, black pepper, and red pepper flakes (if using), whisking well to distribute the seasonings evenly. Bring the soup to a gentle simmer over medium heat, stirring frequently. The soup will gradually thicken as the roux cooks the liquid—this typically takes 10-15 minutes. I always taste and adjust seasoning halfway through this step because the flavors develop as the soup simmers.

Once the soup has thickened and coats the back of a spoon lightly, reduce the heat to low and add the shredded cheddar cheese in three additions, stirring after each handful until completely melted before adding more. Keep the heat low—this prevents the cheese from becoming stringy or separating. Once all the cheese is melted and the soup is smooth and creamy, taste and adjust seasonings if needed. The soup is ready to serve immediately, or you can keep it on low heat for up to 20 minutes before serving.