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Halve 2 onions. Peel and coarsely chop. Peel and smash 4 garlic cloves, then coarsely chop. No need to break out your best knife skills because it’s all going to get blended, but try to cut everything into somewhat even pieces so they cook at the same rate.
Melt 3 Tbsp. unsalted butter in a large Dutch oven or pot over medium-low heat. Add onion, garlic, and ½ tsp. red pepper flakes; season with salt and black pepper. Stir to combine, then cover pot and cook, stirring occasionally, until softened and just starting to take on color, 8–10 minutes.
Meanwhile, trim bottoms of 2 heads of broccoli, then peel. Separate hefty stalks from florets (the thin, little stalks attached to broccoli tops are fine to leave attached to the crowns). Set aside florets. Coarsely chop stalks into 1"–2" pieces. Peel 1 medium russet potato. Coarsely chop into 1"–2" pieces.
Add broccoli stalks and potato pieces to pot and stir to coat in butter. Pour in 5 cups water, add 1 Tbsp. salt, and bring to a boil over high heat. Reduce heat to medium-low and keep at a simmer. Cover pot and cook until broccoli stalks and potatoes are completely tender (test with a cake tester, paring knife, or fork), 20–25 minutes.
Meanwhile, chop reserved florets into small pieces (you can take the time to separate them into baby florets or just have at them with your chef’s knife). Grate 8 oz. sharp cheddar cheese on the large holes of a box grater. You should have about 2 1/2 cups.
Add half of florets to pot, cover, and cook until bright green, 3–5 minutes. Transfer to a blender, let cool for 5 minutes, and purée until smooth. Some blender advice: Don't fill your blender more than halfway—work in batches as necessary. To avoid explosions, remove the center lid from the blender and cover with a kitchen towel. Alternatively, use an immersion blender. Your soup won’t be quite as smooth, but it will still be great. Return purée to pot.
Return to medium heat and add remaining florets. Cover pot and cook until bright green, another 3–5 minutes. Add 2 cups cheddar cheese and stir until smooth. Cover and reduce heat to low; set aside while you make the cheddar croutons.
Heat broiler. Broil 4 thick slices rustic white bread on a rimmed baking sheet until lightly browned (watch closely: broilers vary dramatically in power!), 2–3 minutes. Turn bread over (soft side up), then sprinkle with reserved cheddar cheese and a few flecks of red pepper flakes. Continue to broil until cheese is bubbling and browned in spots, 2–3 more minutes.
Remove soup from heat and stir in ½ cup plain Greek yogurt. Season generously with black pepper; taste for salt.
Cut cheese toasts into 1" (spoon-sized!) pieces. Divide soup among bowls. Top with cheddar croutons.