In a large saucepan, combine Jerusalem artichokes and milk. Bring to a boil, and reduce to a simmer, and cook for 10 minutes. Drain, reserving 3/4 cups of the milk.
Preheat oven to 450 degrees. In a medium bowl, whisk creme fraiche, reserved milk, lemon juice, 1/4 cup Gruyere, thyme, salt, and pepper. Add artichokes, potatoes, chestnuts, shallots; gently mix to combine. Transfer to a 6-cup shallow baking dish, and cover tightly with parchment-lined aluminum foil. Place on a rimmed baking sheet, and bake until artichokes are tender, about 1 hour. Remove foil, and sprinkle with breadcrumbs and remaining 3/4 cup cheese. Bake, uncovered, until golden brown, 8 to 10 minutes. Serve warm.