Heat the oil in a large Dutch oven over medium-high heat until shimmering. Add the squash, carrot, celery, and onion; cook until starting to soften, about 5 minutes. Add the garlic and cook, stirring, until fragrant, about 1 more minute.
Add the rosemary, sage, thyme, bay leaf, broth, and chicken. Bring to a boil (this might take 5-10 minutes), then reduce the heat and gently simmer the soup, uncovered, until the vegetables are tender, about 10-15 minutes.
Stir in the noodles and kale and cook until just barely al dente, about 5 minutes for wide egg noodles. Remove from heat and stir in vinegar. Season with salt and pepper, to taste.