Step 1

Preheat oven to 400ºF. Slice 1/4 inch off top of garlic head to expose cloves. Place, cut side up, in the center of a large piece of foil. Drizzle garlic with 1 tablespoon oil. Bring sides of foil up over garlic and fold tightly to seal; place on a baking sheet. Roast until very fragrant, soft, and golden, 40 to 45 minutes. Unwrap and let stand until cool enough to handle.

Step 2

Heat 2 tablespoons oil in a large saucepan over medium heat. Add onion and cook, stirring occasionally, until very tender and beginning to turn golden, about 5 minutes. Add broccoli, potato, and remaining 1 tablespoon oil; sprinkle with 1/4 teaspoon salt. Cook, stirring, until slightly softened, about 3 minutes. Pour in 5 cups broth. Increase heat to high; bring to a boil. Reduce heat to medium-low; simmer until broccoli and potato are very tender, about 25 minutes. Squeeze roasted garlic cloves out of head; add to soup. Simmer for 5 minutes.

Step 3

Working in batches, puree soup in a blender. (Remove center piece of blender lid to allow steam to escape and cover opening with a towel.) If soup is too thick, stir in up to 1 cup additional broth, 2 tablespoons at a time. Stir in vinegar, pepper, remaining 3/4 teaspoon salt, and honey, if desired. Top with yogurt, pepper, and parsley, if desired.