Put eggs in a small saucepan filled with water. Bring to a boil, cover, and remove from heat. Let sit 15 minutes, then cool, peel, and halve.
Heat 1 tbsp. oil in a large frying pan over high heat. Add chicken, skin side down, and cook until well browned, about 5 minutes; transfer to a plate.
Add remaining 1 tbsp. oil, spices, salt, and onion and cook, stirring constantly, until fragrant, about 1 minute. Add rice and 2 cups hot water. Place chicken, skin side up, on top of rice and simmer, covered, until chicken is cooked through, 10 to 12 minutes.
Mix yogurt, mint, and cilantro together in a small bowl. Set eggs into pan with chicken and serve with yogurt on the side.
Note: Nutritional analysis is per serving.