Wash and chop the broccoli and grate the carrot. Grate the cheese.
Heat a large heavy-bottom pot over medium heat and add the olive oil.
Add the chopped onion and sauté until soft and translucent. Add the salt, pepper, paprika, garlic powder, and onion powder. Add the butter and let it melt.
Sprinkle in the all purpose flour and mix well so the flour toasts in the melted butter. Add the chicken stock gradually and mix well with a whisk until no lumps remain. Add the bay leaves, the milk, and half and half cream.
Let the soup mixture come to a boil, then add the broccoli florets and carrots and let them cook as the soup simmers for about 4 minutes. You can turn the heat to medium-low once you add the veggies, to keep the soup at a simmer and prevent it from boiling too hard.
When the broccoli is still vibrant green but tender, turn the heat off, remove the bay leaves, and add the cheddar cheese. Mix well until melted.
Serve this soup immediately with some extra shredded cheese on top.