Heat the oil in a large saucepan or skillet set over medium heat. When the pan is hot, add the chicken. Season with the salt and pepper and cook without stirring for 2 minutes.
Add the onions, stir and continue to cook until the chicken is no longer pink in the center, 3 minutes. Transfer to a bowl and cover.
Add the broth, milk, squash, mustard and bay leaf to the saucepan and bring to a boil over medium-high heat. Mash the squash with a potato masher until mostly smooth.
Add the pasta and stir to combine. Cover, reduce heat to low, and cook for 6 minutes, stirring once.
Fold in the chicken-onion mixture and kale with a rubber spatula, cover and cook until the kale has wilted, 2 minutes.
Remove the pan from the heat and stir the cheese and nutritional yeast into the pasta. Serve immediately.
