Prepare the lid: Place the potatoes and parsnips in a medium pot, and cover them with a couple inches of salted water. Bring to a boil, and then simmer over medium heat for 12-15 minutes, until both are easily pierced in the center with a knife. Drain, scoop into a bowl and mash with a fork or potato masher. Add the butter, salt and pepper and stir. Add the crème fraîche and milk and stir to combine. If the potato mixture is thicker than you’d like and would be difficult to spread, add more milk, 1 tablespoon at a time. Taste and adjust the seasoning as needed.
Heat the oven to 400 F.
Cook the vegetables: In a large pan (4-quart, or an 11-inch deep skillet or braiser) heat the olive oil over medium heat. Add the onion, carrots, pinch of salt and pepper, and sauté until the vegetables begin to brown ever so slightly, about 10 minutes. Add the garlic and cook for 1 minute more. Add the fresh mushrooms, a pinch of salt, fresh thyme, and increase the heat to high, cooking the mushrooms until they brown and soften, releasing their juices, about 5-10 minutes. Sprinkle with flour and cook 1 minute.
Add the sherry and scrape up any stuck bits. Add the tomato paste, and stir to combine. Add the stock slowly, stirring (start with 1/2 cup stock and add more if mixture seems to dry). Simmer over medium heat until mushrooms are coated in a thick sauce. Taste and season with additional salt and pepper if desired.
Bake the pie: In an oven-safe 2-3 quart baking dish, scoop the mushrooms into an even layer. Top with potato mixture, using the back of your spoon to spread it evenly over the top, all the way to the edges to form a seal. Bake for 25-30 minutes, until lightly browned on top. Garnish with additional thyme and parsley, if desired.