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Shred/grate the cheese and set aside. Measure out remaining ingredients prior to beginning.
Melt butter in a large soup pot over medium heat and add the onions. Stir to combine and let them soften for 5 minutes.
Add the garlic, Worcestershire sauce, hot sauce, and seasonings and cook for 1 more minute. Add the flour and stir to combine, cook for 1-2 minutes, until the raw flour smell is gone.
Add the chicken broth in small splashes, stirring continuously. Add the half and half in the same manner. Bring to a gentle bubble. Let it simmer, reduce, and thicken for 5-7 minutes, uncovered, stirring occasionally. Run a silicone spatula along the bottom of the pot throughout cooking to ensure nothing settles/sticks to the bottom.
Add the broccoli and carrots. Cover partially and let the broccoli and carrots soften for about 10 minutes, stirring occasionally. Adjust heat as needed to maintain a very gentle bubble.
Once the broccoli is fork tender, reduce heat to low.
Optional: Use an immersion blender and give the soup a quick blend in one spot, then stir to combine so that it’s still got some texture.
Over low heat, gradually stir in the cheese and mix until melted and well-combined. Serve!