Heat oil in large heavy-bottomed sauce pot over high heat until just starting to smoke. Add asparagus pieces and cook, stirring occasionally, until tender-crisp and lightly browned. Season asparagus to taste with salt and pepper, transfer to bowl, and set aside. Do not clean sauce pot.

In a medium saucepan, bring 2 quarts of water and 1 teaspoon kosher salt to a boil over high heat. Add pasta and stir immediately. Allow to return to a boil, stir once vigorously, cover pot, and shut off heat. After one minute, stir one last time. Recover pot. Follow timing on box as a general guide for cooking time—start checking pasta 2 minutes before suggested cooking time and continue checking every minute until proper al dente texture is reached. Pasta should be fully softened, but retain a slight bite in the center. Drain pasta, reserving 2 cups of cooking liquid. Immediately transfer pasta to empty sauce pot from cooking asparagus.

Add asparagus, ricotta, lemon zest, arugula, grated cheese, and a few big twists of black pepper to the pasta. Add 1 cup pasta water and stir until all ingredients are combined and arugula has wilted. Continue adding pasta water until desired consistency is reached (the ricotta should turn into a creamy sauce that coats the noodles). Season to taste with salt and pepper, and serve immediately, passing more grated cheese at the table.