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Step 1
In a large saucepan, melt the butter. Stir in the flour and onions. Cook for about a minute,
Step 2
Slowly add the half-and-half, about 1/4 cup at a time, whisking until thickened and smooth.
Step 3
When all of the half-and-half is incorporated, add the broccoli and processed cheese.
Step 4
When the cheese is completely melted, add the chicken broth 1 cup at a time, stirring occasionally, until soup is well mixed and has a consistent texture.
Step 5
Add the carrots and simmer for about 10 minutes.
Step 6
Stir in the cheddar cheese and cook for 10 minutes more.
Step 7
Season with salt and pepper to taste.
Step 8
Serve when all the cheese is fully melted and blended through.