Cook the rice according to the package directions.
Meanwhile, pat the chicken dry with paper towels. Carefully stuff the ginger under the skin. Season with 1 teaspoon salt and ¼ teaspoon pepper.
Heat the olive oil in a large skillet over medium heat. Add the chicken, skin-side down, and cook until the skin is crisp, 7 minutes.
Turn the chicken and cook, covered, until cooked through, about 6 minutes.
Heat the sesame oil and seeds in another large skillet over medium heat until the seeds are light gold, about 2 minutes.
Add the spinach, ½ teaspoon salt, and 1/8 teaspoon pepper and cook until wilted. Divide among plates and serve with the chicken and rice.