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 in flour from the inner walls of the well until a shaggy dough forms. Knead firmly for 8 to 10 minutes until the dough is smooth, elastic, and springs back when pressed. Wrap tightly in plastic wrap and let rest at room temperature for 30 minutes.
02 - Melt 2 tablespoons of butter in a skillet over medium heat. Add the chopped leeks and sauté for 6 to 8 minutes until softened and translucent, taking care not to brown them. Allow the leeks to cool completely, then gently fold together with the drained burrata, lemon zest, salt, and pepper until evenly combined. Set aside.
03 - Divide the rested dough in half and keep one portion wrapped while working. Roll out each portion on a floured surface or pass through a pasta machine to approximately 1 mm thickness. Cut the sheets into squares about 3 inches per side. 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 little water if needed. Arrange the triangoli 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 fresh sourdough breadcrumbs and toast, stirring frequently, until golden and crispy, about 3 to 4 minutes. Stir in the lemon zest, parsley, and a pinch of salt. Remove from heat and set aside.
05 - Bring a large pot of generously salted water to a gentle boil. Cook the triangoli 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, then add the lemon juice, lemon zest, and a generous amount of freshly cracked 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 lemon butter breadcrumbs, and garnish with additional parsley if desired.