Heat olive oil in a skillet or grill pan over medium-high heat. Meanwhile, place chicken breasts in a sealed plastic bag and with a mallet, pound to ½ inch thickness. Sprinkle the chicken with 1 teaspoon kosher salt.
Cook chicken in the pan for 3 to 4 minutes per side, or until cooked through (165 degrees F on an instant-read thermometer).
Bring a large pot of salted water to a boil and cook pasta according to package directions. Drain when cooked and reserve 1 cup of the cooking water.
Transfer the chicken to a cutting board and when slightly cooled, cut into strips.
Heat the butter in a large pot over medium-high heat. When melted, add the leeks and garlic. Cook until soft (about 5 minutes). Then add the lemon juice, zest, lemon slices, and remaining teaspoon of salt. Cook for another 2 minutes. Then stir in the reserved water.
Add the chicken and spinach to the pot and stir to combine. Let cook until spinach is wilted.
Combine the cooked pasta and the parmesan cheese to the pot and stir to combine.
Serve and top with additional optional parmesan cheese if desired.