Whisk together flour and 1/3 cup chicken broth until smooth.
Combine remaining chicken broth and next 3 ingredients in a Dutch oven. Bring to a boil; cover, reduce heat, and simmer 8 minutes or until potatoes are tender. Gradually stir in flour mixture. Cook, stirring often, 5 minutes.
Stir in milk and 8 ounces shredded cheese. Cook mixture over medium-low heat, stirring constantly, until cheese melts. Top each serving of soup with 1 teaspoon cheese, 1 teaspoon bacon pieces, and 1 teaspoon chopped green onions.
Note: We recommend freshly shredded cheese (versus the preshredded variety) for additional creaminess and even melting.