Bring a large pot of salted water to a boil.

Pat the chicken dry and season with salt and pepper.  Add 1 tbsp of butter and 1 tbsp of olive oil to a large skillet over medium-high heat.  When hot, add half of the chicken and cook until no longer pink and starting to brown.  Remove to a plate and repeat with the remaining butter, olive oil, and chicken.

Add the garlic to the hot skillet and cook until fragrant, about 45 seconds.  Pour in the heavy cream and simmer for about 5 minutes, until the sauce has thickened.  Stir in the lemon zest and juice.  Remove from the heat and cover.

Add the pasta to the boiling water and cook according to the directions on the package.  When done, drain the pasta and return to the pot, reserving ½ cup of the pasta water.

Add in the spinach and stir with the pasta for a couple of minutes so that the spinach starts to wilt.  Add the chicken and any accumulated juices from the plate.  Add the sauce and stir, making sure to coat the pasta, spinach,  and chicken.  Use some of the reserved pasta water if the sauce is too thick.  Sprinkle in the parmesan cheese and stir to combine.  Taste for seasoning and adjust as necessary.