Boil water and cook the pasta to al dente according to the package instructions. Reserve a ¼ cup of starchy pasta water.
Heat the olive oil in a large nonstick skillet over medium-high heat. Season the chicken with salt and pepper on both sides. Add the chicken to the pan and cook for about 5-7 minutes per side, or until cooked through and an internal temperature of 165°. Remove the chicken from the pan. Allow the chicken to rest for 5 minutes before cutting it into thin slices.
With the same nonstick pan over medium-high heat, melt the butter. Add the shallots and garlic and cook until fragrant and translucent, about 3-5 minutes. Add the red pepper flakes and cook for one minute.
Stir the tomato paste into the shallot-garlic mixture. Cook, stirring frequently, until the paste has coated the shallots and garlic, and begins to darken, about 3-5 minutes.
Carefully add the vodka to the pan and stir to incorporate, scraping any brown bits from the bottom of the pan.
Reduce the heat slightly to medium-low. Add the heavy whipping cream, stirring constantly until combined. Stir in the parmesan cheese until incorporated. Season with salt and pepper to taste.
Finally, stir in the cooked pasta, spinach, basil, chicken, and starchy pasta water. Toss until pasta is coated with the sauce and spinach has wilted slightly. Garnish with additional parmesan cheese and basil. Serve immediately.