Heat the butter in a large Dutch oven over medium heat.
Add the onion and celery then cook, stirring occasionally, until softened, about 4-5 minutes.
Add the minced garlic then cook, stirring constantly for 1 minute.
Whisk in the flour until lightly browned, about 2 minutes.
Very slowly whisk in the chicken broth, milk, and heavy cream. then cook, whisking constantly until well incorporated, about 2 minutes.
Add the broccoli and shredded carrot; stir until well combined. Bring to a near boil then reduce to simmer and cook until the broccoli is tender 8-10 minutes.
Puree some of the soup with an immersion blender, if desired.
Gradually add the shredded sharp cheddar while whisking, until completely melted. Taste and season with sea salt and freshly cracked pepper, to taste.
Serve with extra cheese, if desired. Enjoy.