Heat oil in an oven-proof frying pan over medium heat. Add the onion and garlic, and sauté until softened.
Add mushrooms and sauté for 2 minutes.
Add feta and basil, and cook until the feta is melted. Season with salt and pepper, and remove pan from heat.
Cut a lengthwise pocket in the chicken breasts and fill each one with half of the mushroom mixture.
Fry the chicken breast in the pan over medium heat for 2 minutes until browned, and transfer chicken to oven. Keep frying pan with juices to make sauce later. Roast chicken for 10-12 minutes until completely cooked through.