In a medium saucepan, cover the lentils with water and bring to a boil. Cover partially and cook over low heat for 20 minutes, stirring occasionally. Add 1/2 teaspoon of salt and cook until tender, about 5 minutes longer. Drain the lentils and spread them on a large baking sheet to cool.
Preheat the oven to 350°. Spread the hazelnuts in a pie plate and bake for 10 minutes, or until lightly toasted. Transfer the hazelnuts to a kitchen towel and let cool. Vigorously rub the nuts in the towel to remove the skins, then finely chop them.
Squeeze the lemon juice into a large bowl of cold water and add the lemon. Peel the Jerusalem artichokes and drop them into the acidulated water. Using a mandoline, slice the Jerusalem artichokes crosswise 1/8 inch thick. In a large pot of boiling salted water, cook the slices until crisp-tender, about 8 minutes. Drain thoroughly and transfer to a large bowl. Add the lentils, hazelnuts, cream, lemon zest and garlic, season with salt and pepper; toss gently. Transfer to a 10-by-15-inch glass baking dish.
In a medium bowl, toss the bread crumbs with the melted butter and scatter over the gratin. Bake for 30 minutes, or until the gratin is bubbling and golden brown. Let stand at room temperature for 15 minutes, then sprinkle with the parsley and serve.
The gratin can be prepared through Step 3 and refrigerated overnight. Bring to room temperature before baking.