In a large saucepan over medium-high heat, heat together the butter, flour, and chopped onion. Cook, stirring, for about a minute, then pour in about a quarter of the half-and-half and whisk until thickened and smooth. Slowly add the rest of the half-and-half the same way.
When the half-and -half has been incorporated, add the broccoli and velvet. When the cheese is melted, stir in the chicken broth a cup at a time until the soup is well mixed and has a consistent texture. Add the carrots and simmer for about 20 min., During the last 10 min, stir in the shredded Cheddar cheese. When all the cheese is completely melted, season to taste with salt and pepper and serve.
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