In a large saucepan, melt the butter. Stir in the flour and onions. Cook for about a minute, then slowly add the half-and-half, about 1/4 cup at a time, whisking until thickened and smooth. When all of the half-and-half is incorporated, add the broccoli and processed cheese.
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. Add the carrots and simmer for about 10 minutes. Stir in the cheddar cheese and cook for 10 minutes more. Season with salt and pepper to taste. Serve when all the cheese is fully melted and blended through.