Using a food processor, finely chop the ginger, parsley, mint and cilantro. Add 3/4 cup peanut oil, the vinegar, honey, 3/4 teaspoon salt and 1/4 teaspoon pepper and process until blended.
In a grill pan, heat the remaining 2 tablespoons peanut oil over medium-high heat. Season the chicken lightly with salt and pepper and grill, turning once, until golden on both sides, 5 minutes. Transfer to a board to rest for a minute, then cut into thin strips.
Spoon the rice into bowls and top with the chicken and dressing.