In a medium bowl, whisk together the yogurt, curry paste, oil, salt, and pepper. Add the chicken pieces and toss to coat evenly. Cover and refrigerate for 30 minutes to 1 hour. Thread marinated chicken onto four 12-inch metal skewers and set aside.
In a small bowl, whisk together all sauce ingredients until smooth. Set aside.
Preheat the oven to 475°F with the rack in the middle position. Line a rimmed baking sheet with aluminum foil. Toss the halved potatoes with 2 tablespoons oil, ½ teaspoon salt, and ¼ teaspoon pepper. Arrange cut side down around the outer edges of the sheet. Roast until the bottoms are golden brown, about 15 minutes.
Remove the baking sheet from the oven and preheat the broiler. Toss the green beans with the remaining 2 tablespoons oil, ¼ teaspoon salt, and ¼ teaspoon pepper. Spread them in the center of the sheet. Arrange the chicken skewers around the perimeter on top of the potatoes. Return the sheet to the oven and broil until the chicken is lightly charred and reaches 160°F, about 6 to 8 minutes per side. Flip the skewers halfway through.
Transfer the chicken, potatoes, and green beans to a serving platter. Drizzle with the prepared sauce and garnish with additional cilantro, if desired.