Place basil, chopped garlic, and 2 tbsp oil in food processor, pulsing until the basil is chopped, about 30 seconds. Avoid pureeing the mixture.
Cut a pocket int each chicken breast. Put half the basil mixture into each pocket.
Heat 1 teaspoons of oil in a frying pan. Add chicken breasts carefully to avoid the spilling the basil mixture from the pocket. Saute until lightly browned. Turn to brown other side. Remove from frying pan. Put chicken into an oven-proof container. Set aside.
Brown onion in the same pan/oil used to saute the chicken adding another teaspoons of oil if needed. Add soy sauce. Remove onions from pan, add to baking dish, arranging so that the chicken breasts sit on top of the onions.
Add a 2-4 Tbsp water to the frying pan. Stir, scraping to remove any browned bits. Add this to the baking pan.
Cover. Bake at 350 degrees F for 30 minutes. Flip chicken breasts. Cover, return to oven and back 30 minutes more.
Remove from oven. Slice each breast in half. Spoon onions and drippings over meat. Serve.