Cut cabbage into thick slices. Cut each slice in half. Combine cabbage, carrots, mushrooms, green onions, and garlic in a large container.

Heat 2 teaspoons oil in a large skillet over medium heat. Cook burger patties, turning occasionally, until heated through, 8 to 10 minutes. Remove from heat and dice.

Heat remaining oil in a large skillet or wok until it shimmers. Add cabbage mixture; cook and stir over medium-high heat until cabbage begins to take on a transparent look, 7 to 8 minutes. Stir in burger pieces. Cook until flavors combine, 7 to 8 minutes more. Stir in soy sauce and sesame oil for an Asian kick. Reduce heat and continue cooking, turning occasionally, until carrots are tender, 3 to 5 minutes.