Bring a large pot of salted water to a boil. Finely zest 2 of the lemons and set aside. Trim the tops and bottoms off the other 2 lemons and cut lengthwise into quarters; remove seeds. Thinly slice the quarters crosswise into triangles. Blanch the lemon pieces in the boiling water for 2 minutes, then transfer with a slotted spoon to a dish towel. Blot dry. In the boiling water, add pasta and cook until just barely al dente. Drain, reserving 1/2 cup of the pasta cooking water. Meanwhile, in a large skillet, heat 1 tablespoon of the oil over high heat. Add the dried lemon pieces and season with a pinch each of salt and sugar. Cook until the lemons are caramelized and browned at the edges, 3 to 5 minutes. Transfer to a plate. Melt the butter with the remaining oil in the pan over medium heat. Add the chile flakes and zest of both lemons; cook until fragrant. Whisk in the reserved pasta water. Toss in pasta, juice of 1 lemon, cheese, pepper and the remaining salt. Cook until pasta is well coated with sauce. Toss in the caramelized lemon and the celery leaves and parsley if using. Taste and add lemon juice if needed. Serve, topped with a drizzle of oil, more cheese if you like, and a sprinkle of sea salt.