Step 1Heat the oven to 400 degrees. Pierce the eggplants in 2 or 3 places with a sharp knife and place them on a jelly roll pan or in a baking dish. Bake until the flesh is soft and the eggplants have collapsed, about 1 hour. Remove from the oven and cool.
Step 2When the eggplant is cool enough to handle, peel away the skin and coarsely chop the flesh. Put the eggplant in a bowl with the garlic, rosemary and olive oil and stir briskly with a wooden spoon so that the eggplant shreds and breaks apart into chunks but does not become a smooth puree. Stir in the salt, vinegar and lemon juice; taste and adjust seasoning. Gently stir in the diced tomato.
Step 3Cut the baguette into one-half-inch thick slices. Toast in a broiler or on the grill until browned on both sides. Spoon on some of the eggplant mixture and use a vegetable peeler to shave a small slice of Pecorino Romano on top. Serve at room temperature.