Preheat the oven to 400F. Slice the tops off each head of garlic, place in an oven safe dish, drizzle with olive oil, then wrap it in foil. Roast until golden and soft, approximately 40 minutes. Set aside to cool, then peel.
For the soup
In a large pot over medium heat, cook the onion and carrots in olive oil until softened, about 3 minutes.
Pour in the chicken broth, roasted garlic, and seasonings. Bring to a low boil for 10 minutes with the lid on to let the flavors meld.
When it's safe to handle, carefully blend until smooth and creamy, or leave just a few chunks. I like to use my immersion blender but you can also use a regular blender. If it's really hot, don't fully seal the lid.
Separately in a bowl or glass, mix the cream and flour together. Slowly pour it into the soup while stirring to mix it in.
Finally add the cheddar cheese and chicken, stirring until the chicken is warmed through and the cheese is melted and silky. Taste and adjust the salt as needed.
Notes
To Make-Ahead: To make this soup quickly, roast the garlic in advance. Once it's cooled, store it in an air-tight container up to 3 days in advance.
Storage: Store leftovers in an air-tight container for up to 3 days.
Reheating: Reheat on the stovetop in a pan. Add a splash or two of broth if needed.