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Melt 1 tablespoon of butter in a small saucepan. Sauté onion in the butter over medium heat for approximately five minutes while stirring intermittently. Onion should just be beginning to brown.
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Add garlic to the onion mixture and cook for about one more minute. Be careful that garlic doesn't burn. Set onion/garlic mixture aside.
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Using a large pan, melt 4 T. butter. Add flour and cook over medium heat for about 3-5 minutes, whisking constantly. You want mixture to thicken.
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Slowly add vegetable broth to the butter mixture, whisking constantly. Then, slowly whisk half and half into the mixture. Simmer for 15-20 minutes until reduced and thickened. Whisk mixture occasionally.
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Add chopped broccoli, carrots and onion/garlic mixture to the roux. Add salt, pepper, optional paprika, optional dry mustard powder and optional cayenne.
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Simmer soup over low heat for 20-25 minutes or until it has reduced and thickened. Whisk while the soup is simmering. After simmering for 20-25 minutes, add most of the cheese. Stir until cheese has melted and is incorporated fully into the soup
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Garnish soup with reserved grated cheese and smoked paprika.