In a 5- to 6-quart pan, combine onion, ginger, chili, and oil. Stir often over high heat until onion is lightly browned, 3 to 5 minutes. Stir in garam masala. Scrape mixture into a blender; add tomato paste and chicken broth. Whirl until very smooth. Pour mixture back into pan, add cream, and bring to a gentle boil over high heat (mixture is inclined to spatter). Reduce heat and simmer, stirring often, until reduced to 3 cups, about 5 minutes. Pour sauce into a bowl. Rinse and dry pan.
Mix chicken with pepper and sprinkle lightly with salt. Set pan over high heat; add 1 tablespoon butter and the chicken. Stir until chicken is no longer pink on the surface, 2 to 3 minutes. Add the sauce and simmer over medium heat, stirring often, until chicken is no longer pink in the center (cut to test), 3 to 4 minutes. Cut remaining 3 tablespoons butter into chunks and stir into sauce until melted.
Spoon chicken and sauce onto rice. Add salt to taste. Squeeze lime juice over portions.