Heat gas or charcoal grill. Place bell peppers on grill. Cover grill; cook over medium-high heat 10 to 13 minutes, turning every 3 to 4 minutes, until all sides are blistered and charred.
Place peppers in brown paper bag; fold down top. Let stand 5 minutes.
Carefully peel as much skin from peppers as possible. Cut in half lengthwise; remove stems, seeds and ribs. Sprinkle with salt.
Place 1 slice of cheese in each pepper half; drizzle with oil. Return pepper halves to grill; cook about 5 minutes longer or until cheese is melted. Sprinkle with basil. Cut each pepper half in half again.