In a stockpot, melt butter over medium heat. Cook onion, garlic and carrot in butter until softened.
Stir in broccoli, and cover with chicken broth. Bring to boil, the reduce and simmer until broccoli is tender, 10 to 15 minutes. - Do Not Cover!
Stir in cheese cubes, cornstarch, and half and half
Simmer and stir until cheese is melted melted.