In a bowl, combine soy sauce, brown sugar, sesame oil, garlic, and ginger. Add the sliced beef and marinate for at least 30 minutes.

Rinse the jasmine rice under cold water until the water runs clear. In a rice cooker, combine the rice and water, and cook according to the manufacturer's instructions.

Preheat the grill to medium-high heat. Grill the eggplant rounds and bell pepper slices for about 5-7 minutes on each side until tender and slightly charred.

In a skillet over medium-high heat, cook the marinated beef for about 5-7 minutes until browned and cooked through.

In serving bowls, layer the cooked rice, grilled vegetables, and beef. Drizzle with BBQ sauce and top with chopped green onions.