Heat the olive oil in small pot over medium heat with garlic and rosemary to infuse flavor, 3 to 4 minutes.
Brush eggplant with olive oil and season with salt and pepper, grill 2 to 3 minutes on each side to tenderize. Remove from heat.
Make sandwiches on split flat breads of 2 slices each eggplant, blue cheese, prosciutto and a handful of arugula.
Heat a dry skillet over medium heat. Choose a second skillet that can fit down inside the first and get a few heavy cans ready to weight skillets. When pan is hot, add 1 flat bread, weight down with second skillet and cans and cook 1 minute, flip, cook a minute more. Repeat process with remaining ingredients and serve.