1. Warm 2 Tbsp. oil in a large skillet over medium-high heat. Pat chicken dry. Season with salt and pepper. Add to skillet in a single layer. Cook until golden brown on both sides, about 6 minutes total, flipping once or twice. Remove to a plate.
2. Reduce heat to medium and add remaining 2 Tbsp. oil to skillet. Sprinkle flour over and cook, whisking constantly, until mixture is pale golden in color, about 1 1/2 minutes. Whisk in broth and wine, scraping up browned bits from bottom of skillet. Bring to a simmer; add chicken back to skillet.
3. Place 4 or 5 sage leaves on top of each breast half; sprinkle with cheese. Cover pan, reduce heat to medium-low and simmer until chicken is cooked through, about 6 minutes. Serve with sauce.