Place the parsnips, carrots, and sweet potato into a large pot and cover with salted water. Bring to a boil over high heat, then reduce heat to medium-low, cover, and simmer until tender, about 10 minutes. Drain and allow to steam dry for a minute or two.
Melt butter in a large skillet over medium heat. Cook and stir mushrooms and leeks until tender, about 5 minutes. Whisk in the flour, and stir until the mixture becomes paste-like and light golden brown, about 5 minutes. Gradually whisk the vegetable broth into the flour mixture. Stir in thyme, salt, pepper, and hot pepper sauce, then add the drained vegetables. Spoon vegetable mixture into a 10 inch deep dish pie pan.
To prepare biscuit topping, combine biscuit mix with milk in a bowl. Drop heaping tablespoons of batter onto vegetable filling. Do not cover filling completely with batter. Bake pot pie in the preheated over until topping is golden brown, 18 to 20 minutes.