Add a large pot to the stove over medium heat. Add butter, onion, and minced garlic. Saute until translucent, about 6 minutes. Add broccoli, carrot, chicken broth, half and half, salt, black pepper, ground paprika, and bay leaves. Stir and bring to a boil. Simmer uncovered until vegetables have softened, about 10 minutes.
In a small bowl, stir together cornstarch and remaining chicken broth to create a slurry; stir into soup. Bring soup to a boil, stirring occasionally. Cook and stir until thickened. Remove bay leaves. Stir in cheese. Serve and enjoy!