Heat a large cast iron pan or skillet over medium heat, add the olive oil and heat through. Add the onion, carrot, celery and squash, and sauté, stirring occasionally, until the onion is opaque, about 6 minutes.
Add the mushrooms and continue to cook, until they have reduced in size and are golden brown, about an additional 5 minutes.
Next, stir in the shredded Brussels sprouts and continue to sauté until they are just wilted, about an additional 3 to 4 minutes. Stir in the turkey until combined.
Add the butter and melt then stir in the flour, stirring to coat the vegetables in the flour, letting it cook for about 1 minute.
Next, add the chicken stock to the mixture and bring to a boil then reduce to a simmer, stirring occasionally, until the sauce becomes thick, about 5 minutes.
Spread the mixture in a 2-quart baking dish. Evenly spread the mashed potatoes over the turkey and vegetable mixture. Bake until the tips of the mashed potatoes are golden brown, about 30 minutes.
Let cool slightly, for about 5 minutes then garnish with chives and parsley. To serve, ladle into bowls.