Lightly grease 2 large baking sheets; set aside. Prepare hot roll mix according to package directions for basic dough, using the 1 egg and substituting the 2 tablespoons oil for margarine. Stir in 1 tablespoon of the fresh or 1/2 teaspoon of the dried basil.
Knead dough and let rest as directed. Divide dough in half. Roll each portion into a 12- or 13-inch circle. Place on the prepared baking sheets.
In a medium skillet cook onion and the remaining 3 tablespoons fresh or 1-1/2 teaspoons dried basil in the 3 tablespoons hot oil until onion is tender. Using fingertips, press indentations every inch or so in dough rounds. Top dough evenly with onion mixture. Sprinkle with olives and cheese. Cover and let rise in a warm place until nearly double in size (30 to 40 minutes).
Bake in a 375 degree F oven for 15 to 20 minutes or until golden brown. Transfer to wire racks; cool. Makes 16 servings.