In a pan over low/medium heat, add 2 T butter and diced onion. Sauté until the onion is translucent and light golden brown, about 5 minutes.
Add the minced garlic and cook for an additional minute, or until the garlic is light golden.
Turn the heat to low. Add the beer and let it cook for an additional 10 minutes.
In a separate pot over medium heat, whatever one you're going to be cooking the soup in, add 4 T of butter and flour. Combine and cook for 5 minutes, stirring constantly.
Add the chicken broth and half and half, stirring constantly.
Turn the heat to low and let it simmer for 20 minutes, stirring occasionally.
Add the broccoli, carrots, and onion + garlic mixture.
Add the salt, pepper, paprika, and ground mustard.
Let the soup simmer for an additional 20 minutes.
Grate the cheddar cheese into the soup. Stir until it has fully incorporated. This should only take a minute or so.