Place bacon strips in a single layer in a dutch oven or large pot over medium heat. Cook until crispy, 5 to 10 minutes, flipping occasionally. Transfer to a paper towel to drain.
Add diced onions, carrots, and celery to the pot with the bacon drippings. Cook over medium heat until softened, about 10 minutes, stirring frequently.
Add chopped broccoli, heavy cream, and 2 cups of water. Bring to a simmer over medium-high heat and reduce to maintain a simmer. Simmer until the broccoli is tender, 5 to 10 minutes, stirring occasionally.
Slowly add shredded cheddar cheese, stirring in until melted and incorporated into the soup.
Add butter and minced garlic, stirring until the butter is melted. Simmer for about 10 minutes to thicken the soup to your desired consistency.
Taste the soup, and add salt and pepper to taste.
Serve hot in bowls. Sprinkle additional cheddar cheese and crumble bacon on top.