Slice each chicken breast in half lengthwise to make 4 cutlets, then pound each cutlet to an even thickness with a meat mallet or rolling pin.
Season both sides of each cutlet with the kosher salt and black pepper.
Place the flour in a shallow dish and dredge each cutlet in the flour, shaking off excess; set the dredged cutlets aside on a plate.
Heat 1 tablespoon butter and the olive oil in a large skillet over medium heat until the butter is melted and foaming.
Add the chicken cutlets to the skillet and brown until golden, about 4 to 5 minutes per side; transfer the browned chicken to a plate and keep warm.
Reduce the heat to medium-low, add the remaining 1 tablespoon butter to the skillet, then add the minced garlic and sauté for about 30 seconds to 1 minute until fragrant.
Stir in the Italian seasoning, then pour in the chicken broth and scrape the pan bottom to deglaze, loosening any browned bits.
Add the heavy cream and bring the sauce to a gentle simmer, stirring to combine.
Stir in the fresh spinach and freshly grated Parmesan until the spinach wilts and the cheese melts into the sauce.
Return the chicken to the skillet, spoon sauce over the cutlets, and simmer on low until the chicken is heated through, about 2 to 4 minutes.
Taste the sauce and adjust seasoning if needed, then serve the chicken florentine with the sauce spooned over it.
