Preheat oven to 425 degrees. In a small saucepan, heat butter and oil over medium heat. Add garlic and salt; cook, stirring, until garlic is fragrant but not browned, about 1 minute. Remove from heat; set aside.
Sprinkle the bottom of two rimmed baking sheets with cornmeal. Divide the dough into four equal pieces. Roll out each piece into a 1/4-inch-thick round. Arrange two rounds on each sheet. Brush with butter mixture, leaving most of the garlic behind. Sprinkle rounds with chopped rosemary; arrange a few small sprigs on top of each.
Bake flatbreads until golden brown, about 15 minutes. Transfer to a wire rack to cool slightly. Drizzle with oil, and season with salt. Serve warm or at room temperature.