Chop the chicken breast into small, bite-sized pieces.

Heat the olive oil in a large soup pot or Dutch oven over medium heat and cook the chopped chicken breast until cooked through, about 5-7 minutes. Remove it from the pot and set it aside on a clean plate. 

Add the butter and warm it over medium heat until melted.  Once melted, add the onion and garlic and sauté for about 3 minutes, until soft and fragrant.

Stir in the flour and mix until smooth, and no clumps remain.

Pour in the chicken broth.  Whisk continuously over medium heat for 5-7 minutes, until the mixture has thickened.  You will know it has thickened when it starts to coat the sides of the soup pot.

Add the broccoli florets and shredded carrots.  Cook for an additional 2 minutes to soften the broccoli.

Turn the heat down to low and stir in the cooked chicken breast pieces, milk, and heavy cream.  Bring the soup to a gentle simmer without letting it come to a full boil. 

Once simmering, remove the soup from the heat and stir in the shredded cheddar cheese until melted.  Add the salt and pepper, take a taste, and adjust the seasonings as needed.

Serve immediately with a piece of crusty bread or soup crackers, and enjoy!