Mix cabbage and red bell peppers together in a bowl. Mix onion, garlic, and ginger together in a separate bowl.
Heat a wok or large skillet over medium-high heat; add 1/4 cup sesame oil. Cook and stir beef and onion mixture in the hot oil until beef is browned on both sides, 3 to 4 minutes. Add cabbage mixture to beef mixture; cook, stirring quickly, until cabbage wilts, onion begins to brown, and beef is fully cooked, about 5 minutes.
Stir remaining 1/4 cup sesame oil, soy sauce, sugar, and black pepper into beef and cabbage mixture; toss to coat. Mix water and cornstarch together in a bowl until cornstarch is dissolved; pour into beef and cabbage mixture. Cook and stir until sauce reduces and thickens, about 5 minutes.
