For the dough, on a work surface, mound the flour; mix in the salt. Using the bottom of a measuring cup, make a well in the center, leaving at least 1/2 inch flour at the bottom of the well. Slowly add the whole eggs, yolks and EVOO, treating the flour like a bowl. Using a fork, carefully beat the egg mixture, then gradually work the flour from the inner wall of the well into the eggs. Once the dough starts to clump, use your hands to mix together. Form the dough into a ball, using a spray bottle to liberally spray with water as needed to help the dough come together (it’s a very dry dough). When the dough forms a stiff, solid mass, scrape away and discard any dried clumps of flour from the work surface.
Using your hands or a stand mixer or food processor fitted with a dough hook, knead the dough until smooth and firm, 10 to 15 minutes by hand or 8 to 10 minutes using a machine. Form the dough into a ball and wrap tightly in plastic wrap. Let rest at room temperature 30 minutes.
For the filling, in a medium bowl, mix the burrata, 2 tbsp. Parmigiano, 1 tbsp. chopped mint, 1 1/2 tsp. lemon zest, 2 tsp. kosher salt and freshly ground black pepper.
For the triangoli, unwrap the dough and cut into 4 pieces. Working with 1 piece at a time (wrap the other pieces in plastic), shape each into an oval about 1/3 inch thick and run the dough through a pasta roller, starting with the widest setting and gradually working down to the thinnest, to create long, thin sheets.
Using a knife or fluted cutter, cut the pasta sheets into 3-inch squares. Spoon 1 heaping tsp. filling onto the center of 1 square. Using a pastry brush or your fingertip, lightly brush around the filling with water, then fold one corner toward its opposite corner to form a triangle. Gently press around the filling to remove excess air, then press along the edges to seal. Repeat with the remaining dough and filling to form about 3 dozen triangoli, transferring them to a large rimmed baking sheet dusted with semolina flour. Space them slightly apart to prevent them from sticking together.
Bring a large pot of water to a boil. Salt the water, add the pasta and cook, stirring often, until almost al dente, 2 to 3 minutes.
In a large skillet, melt the butter over high. Using a slotted spoon, transfer the triangoli, 1/2 cup pasta cooking water and the squash to the skillet. Cook, swirling the skillet, until the pasta is al dente and the sauce thickens, 2 to 4 minutes. Add 1/4 cup sliced mint, the lemon juice and parsley. Divide among 4 to 6 plates. Top with more Parmigiano, the pistachios and the remaining 1 1/2 tsp. lemon zest.