Cook fusilli in water with the tablespoon salt until just tender. Reserve 1 cup pasta water before draining, then return pasta to the warm pot off the heat.
Meanwhile, toast walnuts in a dry skillet for 3 to 4 minutes, stirring, and let cool slightly. Season chicken with 1/4 teaspoon salt and pepper, heat its oil in the skillet and cook for 4 to 5 minutes per side until at least 165°F.
Rest chicken for 5 minutes. Pulse basil, toasted walnuts, garlic, Parmesan, pesto oil, lemon juice and remaining salt in a food processor until finely chopped and spreadable.
Slice the rested chicken into strips and add to the pasta with the pesto and 1/4 cup reserved warm pasta water. Toss off the heat until evenly coated.
Add more pasta water a tablespoon at a time if the pesto looks thick, then serve immediately without boiling the basil sauce.