Trim scallions and separate the light and dark green parts. Thinly slice the light green and white parts and set aside. Thinly slice the dark green parts; reserve separately for topping.
Melt 1 tablespoon butter in a large pot over medium heat. Add ½ cup onion and the sliced white and light green scallions; cook, stirring often, until softened, about 2 minutes.
Add quartered potatoes, 2 tablespoons flour and 1½ teaspoons each garlic powder and onion powder; stir to coat. Increase heat to high; stir in 6 cups broth, cover, and bring to a boil. Reduce heat to medium; cook, covered, until the potatoes are tender, 10 to 15 minutes.
Reduce heat to medium-low. Stir in broccoli; cover, and cook until tender and bright green, 3 to 5 minutes. Stir in 2 cups chicken, the softened cubed cream cheese, ½ cup cream and ½ teaspoon each salt and pepper. Cook, stirring, until the cream cheese is melted and the chicken is heated through, about 4 minutes.
Divide the soup among 6 bowls. Top with ½ cup Cheddar, ½ cup sour cream, ¼ cup bacon and the reserved scallions.
