01 - Mound the flour on a clean work surface and create a deep well in the center. Add the eggs, olive oil, and salt into the well. Use a fork to gradually draw flour from the inner walls into the wet ingredients until a shaggy dough forms. Knead firmly for 8 to 10 minutes until the dough becomes smooth, silky, and springs back when pressed. Wrap tightly in plastic wrap and rest at room temperature for 30 minutes.
02 - Melt 2 tablespoons of butter in a sauté pan over medium heat. Add the chopped leeks and cook gently for 6 to 8 minutes until softened but without any color. Allow the leeks to cool completely, then combine with the drained burrata, lemon zest, salt, and pepper. Fold gently to keep some texture in the filling. Set aside.
03 - Divide the rested dough in half and keep one portion wrapped. Roll out the dough on a lightly floured surface or pass it through a pasta machine, working down to the thinnest setting until approximately 1 mm thick. Cut into roughly 3-inch squares. Place 1 teaspoon of filling in the center of each square, fold diagonally into a triangle, and press the edges firmly to seal, dabbing with a small amount of water if needed. Arrange on a floured tray and repeat with the remaining dough and filling.
04 - Melt 2 tablespoons of butter in a skillet over medium heat. Add the sourdough breadcrumbs and toast, stirring frequently, until golden and crispy, about 3 to 4 minutes. Remove from heat and stir in the lemon zest, parsley, and a pinch of salt. Set aside.
05 - Bring a large pot of generously salted water to a gentle boil. Drop in the triangoli and cook for 3 to 4 minutes until they float and the pasta is tender but still al dente. Meanwhile, melt ¼ cup of butter in a large skillet over medium heat. Add the lemon juice, lemon zest, and several cracks of black pepper. Using a slotted spoon, transfer the cooked triangoli directly into the lemon butter sauce and toss gently to coat. Plate the triangoli, sprinkle generously with the toasted breadcrumbs, and garnish with additional parsley if desired.