Prepare the chicken. Pat the cutlets dry and season both sides with salt and pepper. Place ¼ cup flour in a shallow dish. Lightly coat each cutlet, shaking off the excess flour.
Sear the chicken. Heat a large skillet over medium-high heat. Add 1 tablespoon olive oil and 1 tablespoon butter. Add two cutlets and cook until golden, about 2–3 minutes per side for thin cutlets, adjusting the heat as needed. Transfer to a plate. Repeat with the remaining oil, butter, and chicken.
Brown the butter with the sage. Reduce the heat to medium-low and let the skillet cool slightly. Remove any blackened bits, leave golden browned bits. Add the butter to the pan Once melted and foaming, add the sage leaves. Cook, stirring frequently, until the butter’s milk solids turn golden brown and smell nutty. Continue immediately with the next step.
Cook the aromatics. Add the shallot and cook over low to medium-low heat until softened, about 2–3 minutes, stirring frequently. Add the garlic and cook for about 30 seconds, until fragrant. If the butter begins darkening too quickly, take the skillet off the heat for a minute or two to cool down.
Deglaze and simmer. Pour in the wine and scrape up the browned bits. Simmer for 1–2 minutes, until slightly reduced. Stir in the broth, heavy cream, and maple syrup. Bring to a gentle simmer and cook for about 5 minutes.
Thicken the sauce. In a small bowl, mash the softened butter and 1 tablespoon flour into a smooth paste. Whisk it into the simmering sauce in small pieces. Simmer gently for 3–5 minutes, stirring, until thickened and the flour is cooked.
Finish the chicken. Reduce the heat to low and stir in the Parmesan until melted. Return the chicken and accumulated juices to the skillet. Gently simmer until the chicken reaches 165°F in the thickest part, checking early to avoid overcooking thin cutlets. Taste and adjust the salt and pepper. Spoon the sauce over the chicken and serve.