2 ½ cups grated sharp white and yellow cheddar cheese, plus more for garnish if needed
Melt butter in a large Dutch oven or pot over medium heat. Add the onion and cook until tender.
Whisk in the flour and cook until golden, then gradually whisk in the half & half until smooth. Add the chicken broth, salt & pepper and bring to a simmer.
Reduce heat to medium-low and cook uncovered, until thickened, about 20 minutes.
Prepare the bread bowls; using a sharp knife, cut a circle at the top of each loaf, leaving a 1” border all the way around. Remove the bread top, then hollow out the middle with a fork or your fingers, leaving a thick bread shell.
Add the broccoli and carrot to the broth mixture and simmer until tender, about 20 minutes.
Puree the soup with an emersion blender (can use a blender, but use it in batches) until smooth. Return to the pot if you used a blender.
Add the cheese to the soup and whisk over medium heat until melted.