Heat a large skillet over medium heat. Add enough olive oil to coat the skillet. Season the chicken breasts generously with salt and pepper. Cook until golden brown on one side, 8 to10 minutes; flip and cook until cooked through and no longer pink, about 5 minutes more. Transfer to a cutting board and slice into strips.
Bring a large pot of water to a boil. Cook the pasta according to the package directions.
Meanwhile, in a food processor, blend together the basil, walnuts, avocados, lemon juice, garlic and olive oil. Add salt and pepper to taste.
Drain the pasta into a colander. Transfer it to a large serving platter or bowl and toss with the pesto. Top with the sliced chicken. Serve immediately.