Cut unpeeled eggplant into 3/4-inch cubes, (you should have about 2 cups) and place in a a colander. Sprinkle lightly with salt and set aside while preparing remaining ingredients.
Cut chicken diagonally across the grain in 1/4-inch slices, making slices as wide as possible. In a large skillet, heat 2 tablespoons of the oil over medium-high heat. Add chicken in two batches and cook in a single layer for 3 minutes a side or until cooked through, sprinkling lightly with salt and pepper. Remove to a platter and cover with foil to keep warm.
Rinse eggplant and pat dry. Add remaining oil to skillet; cook eggplant mushroom and onion for 3 minutes stirring frequently. Stir in tomatoes, pepper, stock, garlic, thyme, and basil. Simmer uncovered for 20 minutes or until vegetables are tender. Season to taste with salt and pepper.
Mound hot cooked rice on 3 warmed dinner plates. Arrange chicken slices on top and spoon vegetable mixture over chicken.