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Add the oil and butter to a large soup pot and place it over medium-high heat. Once the butter is melted, add the broccoli, carrots, and onions. Cook, stirring occasionally, for 7-8 minutes, until the onion is translucent and the broccoli is soft.
Add the minced garlic and flour. Cook, stirring often, for 2 minutes.
Add one cup of the half & half. Stir until smooth. Repeat with the remaining half & half.
Add the broth, salt, garlic powder, paprika, and pepper. Stir well. Bring the liquid to a simmer. Cook 5 minutes.
Add the cheese to the soup. Stir until the cheese is melted and well incorporated.