Wash and dry produce. • Peel and mince or grate ginger and garlic. Trim and halve cucumber lengthwise; thinly slice crosswise into half-moons.
In a small pot, combine rice, 1 1⁄4 cups water (2 1⁄4 cups for 4 servings), and 3⁄4 tsp salt (1 1⁄4 tsp for 4). Bring to a boil, then cover and reduce heat to low. Cook until rice is tender, 15-18 minutes. • Keep covered off heat until ready to serve.
Heat a drizzle of oil in a large pan over medium-high heat. Add carrots, half the ginger, 3⁄4 tsp salt (1 1⁄4 tsp for 4), and pepper. Cook, stirring occasionally, until just tender, 2-4 minutes. • Transfer carrots to a plate.
In a medium bowl, whisk together cornstarch, gochujang, soy sauce, ponzu, honey, sesame oil, 1⁄4 cup water (1⁄2 cup for 4 servings), and 1 tsp sugar (2 tsp for 4). Set aside.
Pat chicken* dry with paper towels; season all over with salt and pepper. • Heat a drizzle of oil in pan used for carrots over medium-high heat. Add chicken in a single layer; cook, stirring occasionally, until browned and cooked through, 4-6 minutes. • Add garlic and remaining ginger; cook, stirring occasionally, until fragrant, 1 minute. • Pour in gochujang mixture; cook, turning chicken to coat, until sauce has slightly thickened, 1-2 minutes. Season with pepper to taste. Remove from heat. TIP: If sauce seems too thick, stir in a splash of water.
In a second medium bowl, combine cucumber and half the vinegar (all for 4 servings). Season with salt and pepper.
Fluff rice with a fork. • Divide rice between bowls; top with chicken, carrots, and cucumber in separate sections. Garnish with half the sesame seeds (all for 4 servings) and serve.