In a medium bowl, combine bread, ham, sage, and 2 tablespoons oil; season with salt and pepper, and set aside.
Lay chicken flat on a work surface. Using a paring knife, cut a 2-inch-long slit in the thick side of each breast half. Insert knife, and pivot, carefully forming a deep pocket without enlarging opening (or piercing through opposite side). Stuff each pocket with bread mixture, packing tightly; season chicken with salt and pepper.
In a large skillet, heat 1 tablespoon oil over medium-high heat. Add chicken; cook until browned on one side, 6 to 8 minutes. Turn chicken over; cover skillet, and reduce heat to medium. Cook until chicken is opaque throughout, 6 to 8 minutes more. Transfer to a plate (reserve skillet); cover with aluminum foil to keep warm.
Heat remaining tablespoon oil in skillet over medium-high; add broccoli and garlic. Cook, tossing, until broccoli is warmed through, 4 to 6 minutes. Remove skillet from heat; stir in vinegar. Serve stuffed chicken with broccoli.