Called shepherd’s pie precisely because it features lamb, any other base is a tad off, but this version full of chicken and vegetables is a bit of a cross between shepherd’s pie and chicken pot pie—with the rich chicken filling of a pot pie and the mashed potato top of a shepherd’s pie. You can add even more vegetables, if you like, such as peas or asparagus in spring, zucchini or green beans in the summer, or more root vegetables in the fall or winter. If you choose to up the veggie quotient, add a bit more broth to keep things saucy.
Preheat an oven to 450°F.
Scrub the potatoes clean, put them in a pot, cover with water, add 1 teaspoon of the salt, and bring to a boil. Cook until the potatoes are tender all the way through. Drain and pass through a ricer (no need to peel first!), or peel and mash. Stir in the buttermilk, 1/2 teaspoon of the remaining salt, and about half of the parmesan. Add more salt to taste if you like. Set aside.
While the potatoes cook, in a medium pan or pot that will hold the chicken, bring the stock and wine to a simmer over medium-high heat. Add the chicken, cover, reduce the heat to low, and simmer until the chicken is barely cooked through, about 8 minutes for breasts and 12 minutes for thighs. Set aside to let cool.
While the potatoes and chicken cook, clean and finely chop the mushrooms. In a wide pan or pot that will hold all the chicken and vegetables, heat the butter or oil over medium-high heat. Add the mushrooms and the remaining 1/2 teaspoon salt. Cook, stirring occasionally until the mushrooms have released their liquid, that liquid has evaporated, and the mushrooms are starting to brown.
While the mushrooms cook, finely chop the celery; peel and chop the carrots; peel and mince the shallots and garlic.
Once the mushrooms are brown, add the celery, carrot, shallot, and garlic. Cook, stirring frequently until the vegetables are softened, about 10 minutes.
While the vegetables cook, lift the chicken out of its cooking liquid (save the liquid!) and shred or chop it.
Once the vegetables are cooked, sprinkle them with the flour. Cook, stirring constantly, for about 2 minutes. Slowly pour in the reserved chicken liquid, stirring to create a smooth sauce and cooking until the sauce thickens slightly, about 2 minutes.
Stir in the chicken and remove from the heat. Mince the thyme and parsley; stir them into the mixture. Add salt and pepper to taste.
Transfer the chicken and vegetables and sauce into 4 to 6 individual baking dishes. Spoon the mashed potatoes evenly on top, spreading them to cover the mixture on each dish. Sprinkle the dishes with the remaining parmesan. Bake until the chicken mixture is bubbling and the potatoes are browned, 30 to 40 minutes.