To make the filling: Heat the olive oil in a large, heavy skillet over moderate heat until hot but not smoking. Stir in the onion and cook, stirring, until softened, about 5 minutes. Stir in the lamb and the parsley, and cook, stirring and breaking up any lumps, until the lamb changes color. Remove from heat and generously season with salt and pepper. Set aside.
To make the pasta: Sift the 2 cups of flour and salt together into a bowl. Make a well in the center and place the egg and 1/2 cup water in the well. Work the dough with your hands or a wooden spoon into a smooth dough. (You might find that you will need more water if you are rolling the pasta by hand and less if you are using a hand-turned pasta machine.) Knead the dough for at least 10 minutes until smooth and elastic. Cover the dough with a towel and let rest for 1 hour. Divide the dough into 4 pieces. Roll out each piece to 1/16-inch thick with a rolling pin or a pasta machine, and cut into 3-inch squares, discarding the uneven edges. Place 1 to 2 teaspoons filling into the center of each square. Moisten the entire outer edge with water, bring the 4 corners together in the center and pinch tightly to seal, then pinch together the 4 straight edges to seal completely (take care with this step, sealing the dumplings thoroughly will ensure that they will not become unsealed in the water and lose the filling). Repeat with remaining pasta and filling. Place the manti on a lightly floured baking sheet in a single layer.
To make the yogurt sauce: Drain the yogurt at room temperature through a very fine mesh strainer over a bowl for 30 minutes and discard the whey. Stir together the yogurt and garlic in a bowl and let stand at warm room temperature until ready to use.
To make the butter sauce: In a small skillet over moderate heat cook the butter until it is a very light brown, remove the skillet from the heat, and stir in the paprika, coarse salt to taste, and the hot red pepper sauce.
In a large kettle of boiling salted water cook the manti, in batches, stirring to keep them from sticking together, about 7 minutes, or until the pressed together edges are al dente (they will the the longest to cook). Remove them when done with a long-handled sieve or a large slotted spoon and keep warm with a little of the warm cooking liquid. Serve the manti very hot in individual bowls, spoon some of the yogurt sauce, then drizzle some butter sauce and garnish with mint leaves. Pass the remaining yogurt sauce at the table.
Drink: Cankaya Kavaklidere.