Cook pasta according to package directions in lightly salted water. Drain and set aside.
Cut chicken breasts into chunks and sprinkle on salt and pepper.
Heat butter and olive oil over high heat in a large skillet. Add chicken chunks in a single layer and do not stir for a minute or two in order to allow the chicken to brown on the first side.
Turn the chicken and brown on the other side. Cook until done, then remove chicken from the skillet. Turn heat to medium. Add garlic and quickly stir to avoid burning. After about 30 seconds, pour in wine and broth, stirring to deglaze the pan. Allow the liquid to bubble up, then continue cooking until it's reduced by at least half (most of the surface of the liquid should be bubbling at this point.)
Turn off the heat. Add spinach, tomatoes, chicken, and cooked pasta to the skillet. Toss to combine; the spinach will wilt as you toss everything. Add plenty of Parmesan shavings and toss to combine.
Serve with extra Parmesan shavings.