Step 1: Heat olive oil in a large pot or Dutch oven over medium heat. Add chopped yellow onion and cook until softened, about 5-7 minutes.
Step 2: Stir in the minced garlic and cook for another minute until fragrant. Be careful not to burn the garlic.
Step 3: Add the chopped broccoli florets and chicken or vegetable broth to the pot. Bring to a simmer, then reduce heat, cover, and cook for 10-12 minutes, or until the broccoli is very tender.
Step 4: Carefully transfer about half to two-thirds of the soup mixture to a blender (or use an immersion blender directly in the pot). Add the milk, Greek yogurt, and cottage cheese. Blend until mostly smooth and creamy. Return the blended portion to the pot if using a regular blender.
Step 5: Remove the pot from the heat. Stir in 1.5 cups of the shredded sharp cheddar cheese until completely melted and smooth. Reserve the remaining 0.5 cup for garnish.
Step 6: Season the soup with salt and black pepper to taste. Ladle into bowls, garnish with the remaining shredded cheddar and optional fresh chives or parsley, and serve hot.