Peel and dice the carrot into 1/4-inch pieces, dice the onion into 1/2-inch pieces, and cut the broccoli florets into bite-sized pieces about 1-1.5 inches. Shred the cheddar cheese (freshly grated cheese melts much more smoothly than pre-shredded). Have the milk and half and half measured and ready. This prep work ensures smooth cooking without interruptions.

Heat the olive oil in a large pot over medium heat. Once shimmering, add the diced onion and carrot. Sauté for 5-6 minutes, stirring occasionally, until the onion becomes translucent and the vegetables begin to soften. This develops the foundational flavors of the soup through gentle caramelization.

Sprinkle the flour over the softened vegetables and stir constantly for 5 minutes over medium heat. This cooking step cooks out the raw flour taste and creates a roux that will thicken the soup and give it a creamy, luxurious texture. The flour will toast slightly and darken, which builds deeper flavor.

Pour in the cold milk and half and half, stirring constantly to prevent lumps from forming. The cold dairy helps the roux thicken evenly. Continue stirring for 2-3 minutes until the mixture is smooth and begins to thicken. Then add the low-sodium chicken broth, stirring well to incorporate. I find that adding the cold dairy first prevents lumps much better than mixing everything at once.

Add the broccoli florets to the pot and bring the soup to a gentle simmer over medium heat. Cook for 15-20 minutes, stirring occasionally, until the broccoli is completely tender and the soup has thickened to your desired consistency. The vegetables will break down slightly, naturally thickening the soup while infusing it with their flavor.

For a creamier, more refined texture, blend the soup using an immersion blender directly in the pot, pulsing until you reach your preferred consistency—I like to leave it slightly chunky with some vegetable pieces for texture. Alternatively, carefully transfer the soup in batches to a blender if you prefer a completely smooth soup. Remove from heat and stir in the freshly shredded cheddar cheese until completely melted and incorporated, about 1-2 minutes. The residual heat will melt the cheese smoothly without breaking the sauce.