Cook rice: Combine rice, 1¼ cups water, and ½ teaspoon salt in a small saucepan; bring to a boil over high heat. Reduce heat to low, cover, and cook until rice is tender and water is absorbed, about 17 minutes. Top cooked rice with peas, and keep covered until ready to serve.
Prep ingredients: Peel and finely chop shallot. Peel and finely chop 1 tablespoon ginger. Peel and finely chop 2 teaspoons garlic. Pick cilantro leaves from stems; finely chop stems, keeping leaves whole. Pat chicken dry and cut into 1-inch pieces; season with salt and pepper.
Brown chicken: Melt 1 tablespoon each of butter and oil in a Dutch oven or medium pot over medium-high heat. Add chicken and cook, without stirring, until golden on one side but not cooked through, about 3 minutes. Transfer chicken to a plate.
Sauté aromatics: Heat 1 tablespoon each of butter and oil to same pot over medium-high. Add chopped shallots and cook, stirring, until golden, 2–3 minutes. Stir in chopped ginger, garlic, cilantro stems, and 1 tablespoon of the curry powder; cook, stirring, until fragrant, about 1 minute. Stir in 2 tablespoons tomato paste and 1 teaspoon sugar, and cook, stirring, 1–2 minutes.
Simmer sauce: Add 1 cup water and a pinch each of salt and pepper to pot; bring to a boil over high heat. Reduce heat to medium, add chicken and any resting juices, and simmer, stirring occasionally, until sauce is thickened and chicken is cooked through, 3–4 minutes. Remove from heat, then stir in yogurt until incorporated. Season to taste with salt and pepper.
Serve: Fluff rice with a fork, incorporating peas. Spoon rice into bowls and top with chicken tikka masala; garnish with whole cilantro leaves and serve.