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Pat the chicken breasts dry with paper towels. Drizzle with 1 tablespoon of olive oil and season generously with 1/2 teaspoon kosher salt and 1/4 teaspoon freshly ground black pepper on both sides.
Preheat an outdoor grill or indoor grill pan to medium-high heat (400-450°F).
While the grill preheats, begin preparing the pesto. Combine the fresh basil leaves, toasted pine nuts, and garlic cloves in the bowl of a food processor. Pulse several times until finely chopped.
Place the seasoned chicken breasts on the preheated grill. Cook for 6-8 minutes per side, or until internal temperature reaches 165°F and juices run clear. Remove chicken from grill and let rest on a cutting board for at least 5 minutes before slicing.
While the chicken cooks, add the grated Parmigiano-Reggiano and Pecorino Sardo cheeses, 1/2 teaspoon kosher salt, and 1/4 teaspoon freshly ground black pepper to the food processor with the basil mixture. With the motor running, slowly drizzle in 1/2 cup extra virgin olive oil through the feed tube until a smooth, emulsified pesto forms. Scrape down the sides as needed. Add a tablespoon or two more olive oil if the pesto is too thick.
Bring a large pot of water to a rolling boil. Add 2 tablespoons of kosher salt to the boiling water, then add the gemelli pasta. Cook according to package directions until al dente, typically 10 to 12 minutes.
Before draining the pasta, reserve about 1 cup of the starchy pasta cooking water. Drain the pasta using a colander.
Slice the rested grilled chicken into bite-sized pieces or strips.
Transfer the drained pasta and sliced chicken back to the large pot. Add the freshly made pesto to the pasta and chicken. Begin with about 3/4 of the prepared pesto and add more to taste. Toss the pasta, chicken, and pesto until evenly coated. If the pasta seems too dry, add a few tablespoons of the reserved pasta cooking water, one at a time, until the desired consistency is achieved.
Serve immediately, garnished with additional grated Parmigiano-Reggiano cheese if desired.

Pat the chicken breasts dry with paper towels. Drizzle with 1 tablespoon of olive oil and season generously with 1/2 teaspoon kosher salt and 1/4 teaspoon freshly ground black pepper on both sides.
Preheat an outdoor grill or indoor grill pan to medium-high heat (400-450°F).
While the grill preheats, begin preparing the pesto. Combine the fresh basil leaves, toasted pine nuts, and garlic cloves in the bowl of a food processor. Pulse several times until finely chopped.
Place the seasoned chicken breasts on the preheated grill. Cook for 6-8 minutes per side, or until internal temperature reaches 165°F and juices run clear. Remove chicken from grill and let rest on a cutting board for at least 5 minutes before slicing.
While the chicken cooks, add the grated Parmigiano-Reggiano and Pecorino Sardo cheeses, 1/2 teaspoon kosher salt, and 1/4 teaspoon freshly ground black pepper to the food processor with the basil mixture. With the motor running, slowly drizzle in 1/2 cup extra virgin olive oil through the feed tube until a smooth, emulsified pesto forms. Scrape down the sides as needed. Add a tablespoon or two more olive oil if the pesto is too thick.
Bring a large pot of water to a rolling boil. Add 2 tablespoons of kosher salt to the boiling water, then add the gemelli pasta. Cook according to package directions until al dente, typically 10 to 12 minutes.
Before draining the pasta, reserve about 1 cup of the starchy pasta cooking water. Drain the pasta using a colander.
Slice the rested grilled chicken into bite-sized pieces or strips.
Transfer the drained pasta and sliced chicken back to the large pot. Add the freshly made pesto to the pasta and chicken. Begin with about 3/4 of the prepared pesto and add more to taste. Toss the pasta, chicken, and pesto until evenly coated. If the pasta seems too dry, add a few tablespoons of the reserved pasta cooking water, one at a time, until the desired consistency is achieved.
Serve immediately, garnished with additional grated Parmigiano-Reggiano cheese if desired.