Cut zucchini into noodles using a spiralizer.
Heat 1 tablespoon olive oil in a large skillet over medium heat. Stir in tomatoes; cook until starting to brown, 6 to 7 minutes. Add garlic; cook and stir for 30 seconds. Toss in zucchini; season with salt, Italian seasoning, and pepper and stir to combine. Add chicken, spinach, peas, lemon juice, and zest. Drizzle with 1 tablespoon olive oil. Cover and simmer until heated through, about 5 minutes.