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. Using a fork, gradually draw flour from the inner walls into the wet ingredients until a shaggy dough forms. Knead by hand for 8 to 10 minutes until the dough is smooth, supple, and springs back when pressed. Wrap tightly in plastic wrap and let rest at room temperature for 30 minutes.
02 - Melt 2 tablespoons of unsalted butter in a sauté pan over medium heat. Add the finely chopped leeks and cook for 6 to 8 minutes, stirring occasionally, until softened and translucent but not browned. Remove from heat and allow to cool completely.
03 - Gently tear the drained burrata into small pieces and combine with the cooled leeks in a mixing bowl. Add the lemon zest, season with salt and freshly ground black pepper, and fold together delicately until evenly distributed. Set aside.
04 - Divide the rested dough in half and keep one portion wrapped. On a lightly floured surface, roll out the dough using a rolling pin or pasta machine to approximately 1/16 inch thickness. Cut into roughly 3-inch squares. Place 1 teaspoon of filling in the center of each square, fold into a triangle, and press the edges firmly to seal, dabbing with a touch of water if needed. Arrange on a floured tray and repeat with the remaining dough and filling.
05 - Melt 2 tablespoons of unsalted butter in a skillet over medium heat. Add the fresh sourdough breadcrumbs and toast, stirring frequently, for 3 to 4 minutes until golden and crispy. Remove from heat and stir in the lemon zest, chopped parsley, and a pinch of salt. Set aside.
06 - Bring a large pot of generously salted water to a gentle boil. Carefully drop the triangoli into the water and cook for 3 to 4 minutes, or until they float to the surface and the pasta is tender but still has a slight bite.
07 - While the pasta cooks, melt ¼ cup unsalted 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 lemon butter breadcrumbs, and garnish with additional parsley if desired. Serve immediately.