Broccoli is used two ways in this kid-friendly dish: it's simmered until tender for the soup base and roasted for the garnish.
Preheat oven to 400 degrees. In a medium saucepan, heat oil over medium-high. Add shallots, half of garlic, broccoli stems, potatoes, and a pinch of salt. Cook, stirring, 3 to 4 minutes. Add 5 cups water; season with salt. Bring to a boil, then reduce heat and simmer until vegetables are tender, about 10 minutes. Add 3 cups broccoli florets; cook until tender, 8 to 10 minutes more. Let cool slightly. Transfer to a blender with 1 cup cheese; puree until smooth. Season with salt, pepper, and lemon juice.
Stir together remaining cheese and garlic. Place bread on one side of a rimmed baking sheet, and remaining broccoli florets on other. Drizzle both with oil; season with salt and pepper. Top bread with cheese mixture. Roast 15 minutes, tossing florets halfway through. Serve soup topped with toasts, florets, and zest.