Preheat oven to 375ºF.
Place chicken, whole onion half and bay leaves in a large pot and cover with cold water. Place pot over medium-high heat and bring up to a simmer. Reduce heat to medium and simmer until chicken has cooked through, about 20 minutes. Remove chicken from pot and let cool. Reserve about 1 cup (a mugful) of the cooking liquid and then discard the remaining.
While chicken cools, place a large Dutch oven over medium-high heat with 2 turns of the pan of EVOO, about 2 tablespoons. Add chopped onion, mushrooms, carrots, celery and parsnips to the pan, season with salt and pepper, and cover. Cook, stirring occasionally, until softened, 7-8 minutes. While the veggies are cooking, remove the chicken breast meat from the bones and shred.