Place the chicken in a large pot and cover with water. Add more water if needed to immerse the chicken fully. Add the onion and salt. Bring to a boil. Reduce heat and simmer uncovered for 1 hour.
Remove the chicken from the pot and transfer to a bowl to cool. Strain the broth through a fine-mesh sieve into a large bowl. You should have about 6 cups of broth. Discard the spent onion and wipe out the pot. Return it to the stovetop. Note: Quickly rinse off the sieve, so you can use it to drain the pasta later.
Once the chicken is cool enough to handle, pick the meat from the bones, discarding the skin and bones. Shred with your fingers or two forks. Set aside.
Prepare the soup:
Heat the olive oil in the pot over medium heat. Add the onion, carrots, celery, and potatoes and cook for 5 minutes. Season lightly with salt.
Meanwhile, add 2 cups of water to a large glass measuring cup. Add 1 ladleful of the hot chicken broth. Slowly whisk in the cream of chicken soup until well combined.
Pour this mixture–along with the remaining chicken broth and the shredded chicken–into the pot. Add both cans of corn, along with all the liquid from the cans. Taste and season with salt and a generous shake of black pepper. Bring to a boil. Reduce the heat and simmer for 30 minutes or until the potatoes and carrots are fork-tender.
Cook the pasta:
Bring a medium pot of lightly salted water to a boil. Add the farfalle and cook until al dente. Drain.
Finish the soup:
Add the farfalle to the pot and simmer for 5 minutes. Taste and season. Turn off the heat.
To serve:
Ladle the soup into bowls and serve immediately with more black pepper on top. Enjoy!