Place cooked bacon in a large stockpot over medium-high heat; add potatoes, chicken broth, carrots, onion, parsley, celery seed, salt, and pepper. Bring mixture to a boil, cover stockpot with a lid, reduce heat to medium-low, and simmer, stirring frequently and slightly mashing potatoes, until potatoes are fully cooked, at least 20 minutes. Add extra chicken broth if needed.
Whisk milk and flour together in a bowl or measuring cup until frothy; slowly pour into soup while constantly stirring. Bring soup back to a simmer and cook until soup is thickened, 3 to 5 minutes.
Remove soup from heat and stir in American cheese until completely melted. Ladle soup into serving bowls and garnish with green onions.