Add the chives, garlic, parmesan, pine nuts, olive oil, lemon juice, salt, and pepper into a small food processor. Pulse everything together until it forms a mostly smooth paste. If bits of chive or nuts stick to the sides, stop and scrape them down with a spatula before pulsing again. Set the finished pesto aside while you cook the pasta.
Fill a large pot with water and stir in the salt, then bring it to a rolling boil. Add your short cut pasta and cook according to the package directions until it reaches al dente. Give it an occasional stir so nothing sticks to the bottom of the pot. Drain the pasta well once it's done cooking.
Divide the drained pasta between plates or bowls while it's still warm. Spoon a generous amount of chive pesto over each portion and toss gently so every piece gets coated. Warm pasta helps the pesto loosen and spread more evenly across the noodles.
Break half a piece of burrata into rough pieces over each plate of pasta. Finish with an extra pinch of sea salt and a few more twists of black pepper if you like. Serve it right away while the pasta is still warm and the burrata is soft.