In a dutch oven or large pot over medium heat, saute garlic for one minute, until fragrant. (You can add a little oil as needed.)
Add the chicken broth, heavy cream, and chopped broccoli. Increase heat to bring to a boil, then reduce heat and simmer for 10-20 minute, until broccoli is tender.
Add the shredded cheddar cheese gradually, stirring constantly, and continue to stir until melted. (Add 1/2 cup (64 g), simmer and stir until it melts fully, then repeat 1/2 cup (64 g) at a time until all the cheese is used up.) Make sure to keep it at a very low simmer and avoid high heat, to prevent seizing. Remove from heat immediately once all the cheese melts
Use a slotted spoon to remove about 1/3 of the broccoli pieces and set aside. (This step is optional, if you want some pieces in your soup at the end. If you want all of the soup pureed, you can leave them in.)
Use an immersion blender to puree the remaining broccoli.
Reduce heat to low. Add the shredded cheddar cheese 1/2 cup at a time, stirring constantly, and continue to stir until melted. Puree again to make it smooth.
Remove from heat. Add the reserved broccoli florets back to the soup.