Start by washing and chopping the broccoli, carrots, and onions. Fresh vegetables give more vibrant flavor and texture than frozen, so I always reach for fresh broccoli and carrots first.
Melt the butter in a large stockpot over medium heat, then add the onions, cooking until soft and fragrant. Add carrots and broccoli and season with salt, pepper, paprika, and a few drops of hot sauce to build layers of flavor.
Pour in the broth and cream then whisk in the mustard until everything is smooth. Let the mixture come up to a gentle simmer so the veggies soften and the flavors meld together beautifully.
Mix the cornstarch with cold water (or broth) to make a slurry, then stir it into the soup. This step is what gives the soup that perfectly creamy thickness without any graininess.
Reduce the heat and add the freshly grated sharp cheddar. Stir until the cheese is just melted into the creamy base. Avoid boiling after adding the cheese to keep the texture smooth and luscious.
Taste and adjust the seasoning with salt and pepper. Keep on low heat until ready to serve so the soup stays warm and delicious.