2 cups diced potatoes
1 cup chopped celery
1 cup chopped carrots
1 small onion, diced
2 quarts water, or as needed to cover
3 cubes chicken bouillon
6 tablespoons butter
6 tablespoons all-purpose flour
¾ cup milk
2 cups shredded Cheddar cheese
Combine broccoli, potatoes, celery, carrots, and onion in a large stockpot; pour enough water over the vegetable mixture to cover. Add bouillon cubes to the pot. Bring the mixture to a simmer over medium heat; cook at a simmer until the vegetables are tender, about 20 minutes. While vegetables simmer, melt butter in a saucepan over medium heat. Stir flour into melted butter to incorporate. Slowly stir milk into the butter mixture, cooking and stirring until you have a thick sauce, 5 to 10 minutes. Reduce heat under vegetables to low. Stir sauce into vegetable mixture to combine with the liquid smoothly. Add cheese in small amounts, allowing each addition to melt into the soup before adding the next. Serve hot.