Vigorously stir together the yeast, 1/2 cup (2.25 ounces) flour, sugar, and warm water. Cover with plastic and sit in a warm place until it begins to bubble - about 15 minutes.
Whisk in the salt, 1 tablespoon olive oil, and egg. Pour into the mixing bowl for a standing mixer, add the flour, and mix on low using a paddle attachment until all of the flour is incorporated. Increase the speed to medium high and beat until the dough begins to pull away from the sides of the bowl, about 5 - 7 minutes. You will need to scrape the bowl a couple of times. Cover with a damp towel and place in a warm spot in the kitchen away from any drafts until the dough doubles in size - about 45 minutes - 1 hour. I like to put mine in the microwave oven with a cup of very hot water to keep the cavity warm.
Punch down and scrape the dough from the sides of the bowl. Turn onto a lightly floured work surface, and shape into a rectangle. Roll out to approximately 10" X 12", and roughly 1/2" thick.
Brush with half of the melted butter, spread the garlic over the surface, and sprinkle on the cheese, and basil.
Starting at a long end, carefully roll evenly into a long log. Using a very sharp smooth knife, slice the roll into 8 rolls. Baste the bottom of a pie plate or baking pan with the remaining butter and arrange the rolls, fitting them tightly together if necessary. Cover and set in a warm place for 15 minutes while they rise.
Heat the oven to 400˚ F. Brush the rolls with the remaining tablespoon olive oil and bake until the tops are golden brown, about 20 minutes. The internal temperature of the rolls should reach 200˚ F. Serve warm.
The cook time does not include the additional approximately 1 hour of rising time.