1. 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 and pith by running your knife down the center of each quarter to shave it off. Thinly slice the quarters crosswise into triangles about ¼ inch wide. Blanch the lemon pieces in the boiling water for 4 minutes, then transfer with a slotted spoon to a dish towel. Blot dry with paper towels.

2. In the boiling water, add pasta and cook until just barely al dente. Drain, reserving ½ cup of the pasta cooking water.

3. Meanwhile, in a large skillet, heat 2 tablespoons of the oil over high heat. Add the dried lemon pieces and season with a pinch each of salt and a large pinch of sugar. Cook until the lemons are caramelized and browned at the edges, 3 to 5 minutes. Transfer to a plate.

4. Melt the butter with the remaining oil in the pan over medium heat. Add the chile flakes and almost all of the lemon zest; cook until fragrant. Whisk in the reserved pasta water.

5. Toss in the pasta and grated parmesan cheese, and garnish with salt and pepper. Cook until pasta is well coated with sauce. Toss in the caramelized lemon, chopped parsley, and arugula. Let the arugula wilt just slightly. Serve, topped with a drizzle of oil, more cheese if you like, the remaining zest, and a sprinkle of sea salt.