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Prep all the ingredients before you begin as the recipe moves along quickly.
In a Dutch oven or soup pot combine the butter and olive oil. Cook over medium-high heat until the butter is melted. Add the diced onion, carrots, salt and pepper. Cook until the onions are just starting to soften, about 5 minutes. Add the flour and stir to coat the vegetables. Continue cooking and stirring until the flour starts to form a fond on the bottom of the pan.
Add the chicken broth and scrape the bottom of the pan to release any browned bits. Pour in the broccoli and bring to a simmer. Reduce the heat to medium-low, cover and simmer for 10 minutes.
Stir in the half and half, cheddar and American cheeses. Continue cooking and stirring until the cheese is melted. Add the chicken, cover and simmer for another 5 to 10 minutes or until heated through. Check the seasoning and add salt and pepper as needed.
Garnish each bowl with shredded cheddar and fresh cracked black pepper. Serve with crust bread for sopping up all the creaminess!