Using a sharp boning or chef’s knife, cut through the joint to separate wing from breast (if you hit bone, you’re in the wrong spot; pull the wing out farther to help you get to the place where the joint meets the socket). Pull off and discard any skin from legs and breasts (removing this skin prevents excess fat from rendering into the broth; no need to discard skin on the wings). Add carrots, celery, and all of the shredded chicken meat to pot and cook until pasta is cooked through and carrots and celery are tender but not mushy, about 5 minutes.