INSTRUCTIONS
1. Roll out the pasta dough and cut into tagliatelle. Let dry for 30 minutes or up to 3 hours.
2. To make the pesto, in a food processor, combine the cilantro, garlic, pine nuts, olive oil, lime juice, the ½ tsp. sea salt and a few grinds of pepper and process to a paste, stopping occasionally to scrape down the sides of the work bowl. Be careful not to overprocess; the consistency of the pesto should be thick and fluid and have a slightly grainy texture. Transfer the pesto to a large serving bowl.
3. Pat the chicken breasts dry with paper towels, then season on both sides with sea salt and pepper. In a large nonstick fry pan over medium-high heat, warm the olive oil until very hot but not smoking. Working in batches if needed to avoid crowding the pan, add the chicken and cook, turning once, until golden on both sides and just opaque in the center when cut into with a knife, about 2 minutes per side. Transfer to a plate, cover loosely with aluminum foil and set aside in a warm place. Just before you add the pasta to the boiling water, using a sharp chef's knife, cut each breast in half lengthwise, then cut the halves crosswise into thin slices. Squeeze the lime quarter over the chicken and re-cover the chicken.
4. In a large pot over high heat, bring 5 quarts water to a rapid boil. Add the kosher salt and pasta and cover the pot. When the water returns to a boil, uncover and cook for about 5 seconds and then drain, reserving about ½ cup of the cooking water. Blend a few tablespoons of the cooking water into the pesto to make a creamy consistency, then add the pasta to the bowl and stir and toss with the pesto until the strands are well coated with the sauce, adding a little more of the cooking water if needed to loosen the sauce. Arrange the chicken slices on top of the pasta and serve immediately.
