Slice each chicken breast in half lengthwise to make cutlets, then place between plastic wrap and pound with a meat mallet or rolling pin until evenly thick.
Season both sides of each cutlet with the kosher salt and black pepper.
Place the flour on a plate and dredge each seasoned cutlet in the flour, shaking off excess; set aside.
Heat 1 tablespoon butter and the olive oil in a large skillet over medium heat until the butter melts and foams.
Add the chicken cutlets to the skillet (work in batches if needed to avoid crowding) and brown 4–5 minutes per side until golden and cooked through; transfer to a plate and keep warm.
Reduce heat to medium-low, add the remaining 1 tablespoon butter to the skillet, then add the minced garlic and sauté about 1–2 minutes until fragrant. Stir in the Italian seasoning.
Pour the chicken broth into the skillet and scrape up any browned bits from the bottom to deglaze the pan.
Stir in the heavy cream and bring the sauce to a gentle simmer.
Add the fresh spinach and grated Parmesan to the sauce and stir until the spinach wilts and the cheese is incorporated.
Return the browned chicken to the skillet, spoon sauce over the cutlets, and simmer on low for a few minutes until the chicken is heated through and the sauce thickens slightly.
Serve the chicken Florentine with the sauce spooned over; suggested accompaniments are pasta, mashed potatoes, or rice.