Dice the onion, celery, and carrot into uniform 1/4-inch pieces and set aside—this ensures even cooking and helps build a smooth, cohesive soup base. Grate both the sharp cheddar and extra sharp cheddar cheese together and set aside in a separate bowl; freshly grated cheese melts much more smoothly than pre-shredded varieties because it lacks anti-caking agents. Have your vegetable stock and half and half measured and ready. Melt the butter in a large pot over medium heat, then add the diced onion, celery, and carrot. Sauté for 3-4 minutes, stirring occasionally, until the vegetables soften slightly and become fragrant—this early cooking step begins breaking down the vegetables' cell walls, which helps create a naturally thickened, silky soup.

Sprinkle the flour over the softened vegetables and stir constantly for 3-4 minutes. This cooking step is crucial—it eliminates the raw flour taste and allows the flour to fully absorb the butter's fat, which prevents lumps when you add the liquid. The mixture will look paste-like and smell toasty when ready. Slowly pour in the vegetable stock and half and half while stirring continuously to break up any lumps. Keep stirring for 1-2 minutes until the mixture becomes smooth and uniform in color and texture.

Add the kosher salt, black pepper, garlic powder, and cayenne pepper (if using). I find that cayenne adds a subtle warmth and complexity that deepens the cheese flavor without making the soup spicy—just a pinch is perfect. Stir well to distribute the seasonings evenly, then reduce heat to low and simmer for 10 minutes. This gentle cooking allows the vegetables to fully soften and their flavors to meld into the base, creating a more developed, cohesive taste.

Add the frozen broccoli florets directly to the pot—there's no need to thaw them first, as they'll cook perfectly in the simmering soup. Stir to distribute the broccoli evenly throughout the base, then maintain a low simmer for 10 minutes until the broccoli is tender but still has a slight bite to it. Cooking on low heat rather than high prevents the broccoli from becoming mushy and helps preserve its bright, fresh flavor.

Remove the pot from heat and add the grated cheddar cheese mixture from Step 1, stirring constantly until completely melted and incorporated. The residual heat of the soup is usually enough to melt the cheese smoothly without scorching it—if you add cheese directly over active heat, it can become grainy and separated. Taste the soup and adjust seasoning as needed: add a pinch more salt if it needs depth, or a touch more cayenne if you want slightly more warmth. Serve hot.