In a large pot, heat the oil and butter over medium heat. Add the celery, onion, and shallot; cook for 5 minutes, stirring occasionally.
Add the garlic, stir, and cook for another 2 minutes. Stir in the broth, salt, and carrot; bring to a boil, then reduce the heat and simmer uncovered for 25 minutes.
Using a slotted spoon, remove the vegetables and transfer them to a blender with about 1 cup of the broth. Blend until smooth and set aside.
In the pot with the remaining broth, add the noodles and bring to a boil. Reduce to a simmer and cook for 1 minute less than the package instructions. Turn off the heat.
Return the blended vegetables to the pot along with the chopped chicken. Mix well and let the flavors meld for 5–10 minutes.
Serve topped with fresh thyme and grated Parmesan (it’s delicious even without, but these elevate it).