How to Make It
Preheat the oven to 375°. Pierce the sweet potatoes all over with a fork. Place the sweet potatoes on a rimmed baking sheet and bake for about 1 hour, until tender. Let cool slightly.
Meanwhile, fill a medium bowl with water and squeeze 1/2 of the lemon into it. Peel the salsify, adding them to the bowl as you go. Cut the salsify into 1 1/2-by-1/4-inch sticks and return them to the water.
Peel the sweet potatoes and transfer them to a food processor. Puree until smooth. Transfer the puree to a large bowl and stir in the eggs and 1 1/2 teaspoons of kosher salt. Add the 2 cups of flour, a generous pinch of pepper and the nutmeg and stir until a soft dough forms.
Transfer the dough to a heavily floured work surface and gently knead in more flour—as much as 1/2 cup—until the dough comes together but is still very soft. Divide the dough into 4 pieces and roll each piece into a 1-inch-thick rope. Cut the ropes into 1-inch lengths and dust each piece with flour. Roll each gnocchi against the tines of a floured fork to make small indentations. Transfer the gnocchi to a lightly floured baking sheet.
Bring a large pot of salted water to a boil. Add half of the sweet potato gnocchi and cook until they float to the surface, about 1 minute, then cook for 2 minutes longer. Using a slotted spoon, transfer the gnocchi to a large plate. Repeat with the remaining gnocchi.
Meanwhile, bring a medium saucepan of salted water to a boil. Drain the salsify sticks. Add the remaining lemon half and the drained salsify to the saucepan and cook over moderate heat until tender, about 15 minutes. Drain the salsify and discard the lemon half.
In a very large, deep skillet, melt 1 tablespoon of the butter in the oil. Add the shallots and cook over moderate heat until softened, about 2 minutes. Add the salsify, chestnuts and 1 cup of the Ham Hock Stock and simmer over moderate heat until slightly reduced, about 5 minutes. Add the gnocchi and the remaining ham stock and 4 tablespoons of butter and cook, stirring gently, until the gnocchi are hot and the sauce is thickened, about 3 minutes. Season with salt and pepper and transfer to a large platter. Sprinkle with the ham and chives and serve.