In a large pot, combine the broccoli, potato and garlic. Stir in the broth and half of the scallions. Bring to a boil, then lower the heat, cover and simmer until the vegetables are tender, about 35 minutes.
Uncover the soup and stir in the spinach and 1 tbsp. thyme. Return to a simmer. Remove from the heat. Using a blender and working in batches, puree the soup until smooth.
In a small bowl, stir together the yogurt, remaining scallions and 1 tsp. thyme; season with pepper. Return the soup to a simmer, stirring, over medium heat. Ladle the soup into bowls; top with the herbed yogurt.