Heat butter in a large Dutch oven over medium-high heat until melted. Add onion and saute until soft, about 5 minutes, stirring occasionally. Add carrots and continue to cook. Stir in the chicken bouillon cube and saute for an additional minute. Add the flour in and stir frequently, you don't want the flour to burn.
Slowly pour in the chicken stock and using a whisk, thoroughly mix the flour, onion, carrot mixture in with the chicken stock. Once combined, slowly pour in the milk until combined. Continue cooking the soup until it reaches a simmer.
Once simmering, add your cheese and broccoli, stir until completely combined and broccoli florets are tender. Add salt and pepper to taste.
Remove from heat and serve warm.