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In a small saucepan, add 1 tablespoon butter, the diced onion, and sauté over medium heat until the onion is translucent and barely browned, about 4 minutes.
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Add the garlic and cook about 30 seconds, stirring constantly so it doesn't burn. Remove from heat and set pan aside.
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In a large heavy-bottom pot add 4 tablespoons butter, flour, and cook over medium heat for about 3 to 5 minutes, whisking constantly, until flour is thickened.
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Slowly add the stock, whisking constantly.
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Slowly add the half-and-half, whisking constantly.
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Allow mixture to simmer over low heat for about 15 to 20 minutes, or until it has reduced and thickened some. Whisk intermittently to re-incorporate the 'skin' that inevitably forms, this is normal.
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While mixture is simmering, chop the broccoli and carrots. After simmering 15 to 20 minutes, add the broccoli, carrots, and the onion and garlic you previously set aside.
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Add the salt, pepper, optional paprika, optional dry mustard powder.
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Allow soup to simmer over low heat for about 20 to 25 minutes, or until it has reduced and thickened some. Whisk intermittently to re-incorporate the 'skin' that inevitably forms, this is normal.
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While soup simmers, grate the cheese. Stir in the cheese until melted and incorporated fully, less than 1 minute.
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Serve when completely melted.