Heat 3 T. oil in 12 inch non stick skillet over medium-high heat until just smoking. Add eggplant and cook, stirring frequently until tender and browned, about 10 minutes. Stir in 1 T. soy sauce and transfer to plate.
Heat remaining 1 T. oil in now-empty skillet over medium-high heat until shimmering. Add beef, jalapeno, garlic, and ginger and cook, breaking up meat with wooden spoon and stirring occasionally, until browned, about 4 minutes.
Stir in oyster sauce, vinegar, and remaining 1 T. soy sauce. Return eggplant to skillet with beef and continue to cook until combined and slightly thickened, about 2 minutes. Stir in half of scallions. Serve, sprinkled with remaining scallions.