Cook fusilli with tablespoon salt until just tender. Reserve 1 cup water; drain and return to warm pot off heat.
Meanwhile, dry-skillet toast walnuts 3 to 4 minutes, stirring; cool slightly. Season chicken with 1/4 teaspoon salt and pepper. Heat chicken oil in skillet; cook 4 to 5 minutes per side to at least 165°F.
Rest chicken 5 minutes. Food-process basil, walnuts, garlic, Parmesan, pesto oil, lemon juice, and remaining salt until finely chopped and spreadable.
Slice chicken into strips; add to pasta with pesto and 1/4 cup reserved warm water. Toss off heat to coat evenly.
Add water by tablespoon if pesto thick; serve immediately without boiling sauce.