How to Make It
In a medium saucepan, melt the butter in the olive oil. Add the sliced onions and a large pinch of salt. Cover and cook over moderate heat, stirring occasionally, until the onions are very soft and golden, about 20 minutes.
Meanwhile, put the potato slices in a large saucepan of water, add a large pinch of salt and bring to a boil. Simmer over moderately high heat until the potatoes are just tender, about 5 minutes. Drain the potatoes and spread the slices in a single layer on a baking sheet; let cool to room temperature. Gently pat the potato slices dry.
In a large nonstick skillet, heat the duck fat. Add the potato slices and cook over moderately high heat until they are browned and crisp, about 6 minutes on each side. Add the chopped garlic and shake it in the skillet until just golden, about 30 seconds. Add the cooked onions and season them with salt and black pepper. Gently stir in the crushed red pepper and lemon juice. Transfer the potatoes and onions to a platter, sprinkle with the chopped parsley and serve right away.