Heat 1/3 cup vegetable oil in heavy small skillet over medium-high heat. Using tongs, add 1 tortilla and cook until softened, turning once, about 15 seconds for each side. Transfer tortilla to paper towels and drain well. Repeat with remaining tortillas.
Heat remaining 1 tbs vegetable oil in heavy large skillet. Add onion, pepper, oregano and basil and sauté until onion and pepper are tender, about 5 minutes. Season to taste with salt and pepper.
Lightly oil 13x9x2-inch glass baking dish. Spoon 1/2 cup enchilada sauce into dish. Place scant 1/4 cup chicken in center of 1 tortilla. Sprinkle with 1 generous tbs onion mixture. Set aside 1/2 cup cheese for topping. Spoon 2 generous tbs cheese atop chicken. Roll up tortilla and place seam side down in prepared dish. Repeat with remaining tortillas, chicken, onion mixture and cheese. Pour remaining sauce over enchiladas. Sprinkle with reserved 1/2 cup cheese. Cover with foil. (Can be made 1 day
ahead. Chill.) Preheat oven to 350°F. Bake enchiladas, covered, until sauce bubbles and cheese melts, about 35 minutes. Serve hot with sour cream.