Start by chopping the broccoli into small, uniform pieces. Set aside.
Dice the onion finely and mince the garlic cloves. These will form the savory base of your soup.
In a large heavy-bottomed pot, melt the butter over medium heat until it begins to shimmer and smell nutty.
Add the diced onion to the pot and sauté for about 5 minutes until it turns translucent and fragrant, with a slight golden hue.
Stir in the minced garlic and cook for another 30 seconds until it releases a fragrant aroma.
Sprinkle the flour over the onion and garlic, stirring constantly for about 1-2 minutes until the mixture becomes paste-like and slightly golden, helping to thicken the soup.
Gradually pour in the broth while whisking vigorously to prevent lumps, then bring the mixture to a gentle simmer.
Add the chopped broccoli to the simmering broth, then cover and cook for about 8-10 minutes until the broccoli is tender but still vibrant green.
Stir in the shredded cheddar cheese gradually, allowing each addition to melt completely and create a creamy, cheesy base.
Remove the soup from heat and use an immersion blender or transfer carefully in batches to a blender. Puree until smooth and velvety, then return to the pot if using a standard blender.
Stir in the heavy cream, then warm the soup gently over low heat for 5 minutes until heated through. Taste and season with salt and pepper as needed.
Ladle the hot, cheesy broccoli soup into bowls, garnish with extra cheese if desired, and enjoy the comforting, velvety goodness.
