Bring a large pot of water to a boil and salt it generously. Add the pasta and cook until al dente, 8 to 10 minutes. Reserve 1/4 cup of the cooking liquid. Drain the pasta and set aside.
Sprinkle the chicken with salt and pepper. Add 2 tablespoons olive oil to a large pan over medium-high heat. Cook the chicken until the internal temperature reaches 160 degrees F, about 5 minutes per side. Remove the chicken and slice it into strips.
Add 1 tablespoon olive oil to the pan. Add the garlic and shallot and sprinkle with salt and pepper. Cook until fragrant, 1 minute. Add the lemon juice and sugar snap peas and cook, 2 minutes. Add the reserved pasta water and the frozen peas. Cook until the peas are just cooked through, 2 minutes more.
Add the cooked pasta, spinach, sliced chicken, chopped basil, chopped parsley and the remaining 2 tablespoons olive oil. Toss all the ingredients together to coat, season with additional salt and pepper and garnish with parsley leaves. Serve immediately.