Pat chicken breasts dry with paper towel. Salt and pepper them. Heat a large sauté pan over a medium high flame, add half the butter and olive oil and swirl the pan to combine. Add chicken and brown on both sides, 4 or 5 minutes per side. Transfer to plate.
Reduce heat to medium and add remaining butter and olive oil to pan. Add onion and cook ’til softened, about 5 minutes, stirring often to avoid browning. Stir in garlic and cook until fragrant, about 45 seconds. Add diced tomatoes and biryani paste, stirring thoroughly to combine. Return chicken breasts to pan along with any juices. Cover and cook until chicken is cooked through, about 5 to 7 minutes. Turn off heat.
Transfer chicken breasts to cutting board, one at a time, and slice across the grain into 3/4-inch slices. Arrange on plates with Coconut Rice Pilaf. Spoon tomato sauce over chicken—and rice, if desired.
Meanwhile, cook the rice. Combine rice, water, coconut milk and salt in a sauce pan. Bring to boil over high heat. Stir in cumin, reduce heat to low and cover pan. Cook until rice is done, about 15 minutes. Stir in currants, rice vinegar and cayenne pepper. Serve.