1 pound Yukon Gold potatoes, peeled and cut into 1-inch dice
1 pound parsnips, peeled and cut into 1-inch dice
In a small heatproof bowl, cover the porcini with the boiling water and let stand until softened, about 20 minutes. Drain the porcini, straining and reserving the soaking liquid. Coarsely chop the porcini.
Preheat the oven to 400°. Season the chicken with salt and pepper. In a large, deep skillet, heat the olive oil until shimmering. Add the chicken and cook over moderately high heat, turning once, until browned, about 8 minutes. Transfer the chicken to a plate.
In the same skillet, melt 2 tablespoons of the butter. Add the white mushrooms and cook over moderately high heat until browned in spots, about 5 minutes. Add the onion, carrots and celery and cook over moderate heat until tender, about 5 minutes. Add the porcini and season with salt and pepper. Sprinkle in the flour and cook, stirring, for 2 minutes. Return the chicken and any juices to the skillet. Stir in the chicken stock, thyme sprigs and the porcini soaking liquid and bring to a simmer over moderate heat, scraping up any bits from the bottom of the skillet. Simmer until the sauce has thickened and the chicken is cooked through, about 10 minutes. Spoon the chicken into a medium baking dish; discard the thyme sprigs.
Meanwhile, in a large saucepan, cover the potatoes and parsnips with cold water and bring to a boil. Cook over moderately high heat until very tender, about 15 minutes. Drain and return the vegetables to the pot. Cook over low heat until they look dry, about 1 minute. Remove from the heat. Add the remaining 4 tablespoons of butter and mash the vegetables until smooth. Season with salt and pepper.
Spoon the parsnip mash over the chicken and smooth the surface. Bake for 20 minutes, until the sauce is bubbly. Turn on the broiler. Broil the shepherd's pie 4 inches from the heat for about 2 minutes, until browned in spots. Serve piping hot.