Pound the chicken breasts to an even thickness using a meat mallet. Season both sides with salt and pepper. Heat the olive oil in a large sauté pan over medium-high heat. Add the chicken breasts and cook for 3-4 minutes per side until golden brown and fully cooked. Remove the chicken from the pan and set aside.
In the same pan, sauté the minced garlic for 1 minute until fragrant. Add the chicken broth, lemon juice, and lemon zest, scraping up any browned bits from the bottom of the pan. Bring to a boil, then reduce heat to low and stir in the heavy cream. Simmer for 5 minutes, stirring occasionally, until the sauce slightly thickens. Stir in the Parmesan cheese until melted and combined.
Add the fresh spinach to the sauce and cook for about 1 minute until wilted. Return the cooked chicken to the pan, spooning the sauce over the top. Sprinkle additional Parmesan cheese over the chicken and spinach. Cover the pan and cook for another 2 minutes to allow the cheese to melt and the spinach to become tender.
Garnish with thinly sliced lemon and extra Parmesan cheese if desired. Serve immediately.