Heat vegetable oil in a large pot over medium heat. Add onion; cook and stir until soft, 5 to 7 minutes. Drain as much liquid from the onion as possible; transfer onion into a slow cooker.

Add broccoli, celery soup, and condensed soup to the slow cooker. Fill the condensed soup can twice with milk, and pour it into the slow cooker. Add American cheese, and stir to combine.

Cover and cook on Low until broccoli is tender and flavors meld, 3 to 4 hours.