Cook the pasta: Bring a large pot of salted water to a boil. Add pasta and cook until al dente.
Scoop out 3/4 cup of the starchy water before draining.
Make the pesto: In a food processor, blend basil, spinach, nuts, garlic, Parmesan, salt, and pepper. Stream in olive oil, then add water a tablespoon at a time until smooth and spoonable.
Season the chicken: Pat the chicken dry. Toss with salt, pepper, and garlic powder.
Sear the chicken: Heat olive oil in a large skillet over medium-high.
Add chicken in a single layer and cook 3–4 minutes per side until golden and cooked through. Remove to a plate.
Sauté the veggies: In the same pan, add cherry tomatoes with a pinch of salt. Cook 2–3 minutes until they start to burst.
Stir in spinach just until wilted.
Bring it together: Add cooked pasta and chicken to the skillet. Spoon in the pesto and 1/2 cup hot pasta water. Toss over low heat until the sauce turns silky and coats everything.
Add more water if needed.
Brighten and finish: Stir in lemon zest, a squeeze of lemon juice, and Parmesan. Taste and adjust salt and pepper.
Serve: Plate with extra Parmesan and freshly cracked pepper. A few basil leaves on top never hurt.