Preheat oven to 425F. Place broccoli florets, chopped onion, and garlic onto a baking sheet. Drizzle with 1–2 tablespoons of oil and sprinkle generously with salt and pepper. Toss to combine.
Place pan into the preheated oven for 15–20 minutes, just until broccoli is lightly browned.
Drizzle remaining tablespoon of oil into a large dutch oven or pot. Add potatoes and sauté for 2–3 minutes. Stir in the roasted broccoli, onions, garlic from the pan and enough broth to cover, about 2–3 cups.
Bring soup to a boil over medium-high heat and then reduce heat to medium-low, cover, and simmer until potatoes are fork tender, about 10–15 minutes. Turn off the stove.
Add spinach and stir to wilt. Season with a generous pinch of salt and pepper.
Place soup mixture into a high powered blender or use an immersion blender to blend until creamy. Add additional broth, if necessary, to blend. Stir in lemon juice, to taste.
Pour soup into bowls and swirl in plain yogurt and a drizzle of olive oil. Makes 2–3 servings.