Separate the broccoli florets from the stalks. Trim any tough parts from the broccoli stalks then chop tender stalks into small pieces; you should have about 2 cups. Chop the broccoli tops into small pieces.

Heat oil and butter in a large soup pot over medium heat. Add onion and cook until softened, about 5 minutes. Add garlic and cook for an additional minute.

Add broccoli stalks, potato, water, salt, and pepper to the pot. Bring to a simmer then cover and cook until the potato will easily mash when pressed, about 15 to 18 minutes.

Remove soup from heat and stir in yogurt.

Using an immersion blender, blend soup until smooth. Alternatively, use a regular blender and blend soup in batches then pour soup back into the pot.

Stir chopped broccoli florets into the soup and cook until tender, about 8 to 10 minutes.

Stir in 1 cup of the cheese then carefully taste soup for seasoning and add additional salt and pepper as needed. Ladle soup into bowls and garnish with the remaining cheese.