Preheat oven to 350º. For the spinach filling, saute bacon in a large oven-safe skillet until crispy. Set cooked bacon aside to drain on paper towels. Pour off and reserve most of the rendered fat.
In the same skillet, saute the garlic until just starting to brown. Add spinach and stir often, scraping the browned bits off the bottom of the pan, until cooked down significantly and most of the moisture has evaporated, about 10 minutes. Add cooked bacon to the spinach mixture and combine.
Butterfly each chicken breast by slicing nearly through each piece, starting at the long side. Open the chicken pieces like a book and spread the filling evenly across each piece. Fold the pieces back together.
In the empty skillet, heat several tablespoons of the reserved bacon fat over medium-high heat. Carefully add the chicken breasts and sprinkle with salt and pepper, to your taste. Allow the chicken to cook for a few minutes on each side, until golden brown, about 4-6 minutes per side.
Transfer skillet to the oven, and bake until chicken is fully cooked, about 15-20 minutes.
While the chicken is baking, cook the pasta according to the package directions. Once cooked, reserve about a cup of the cooking water before draining.
Return the pasta to its cooking pot (remove it from the heat), add the cheese and a few tablespoons of the reserved pasta water. Stir to combine as the cheese melts, adding more of the pasta water a little at a time, as needed, until you have a creamy sauce.
Once the chicken is fully cooked, remove pan from the oven and allow to rest 5 minutes. Slice chicken and serve on top of a bed of cheesy pasta.