Slice away the eggplants’ green tops. Wash the eggplants and cut them lengthwise into slices ½inch thick.
Set a pasta colander over a bowl or deep dish; line it inside with one layer of eggplant slices, placing them vertically, and sprinkle lightly with salt; place another layer of eggplant over the first and sprinkle a tiny bit more salt over it; and proceed thus until you have used up all the slices. Let stand for at least 30 minutes to allow the eggplant to shed some of its astringent liquid.
Pour enough vegetable oil into a 12-inch skillet to come 1inch up the sides of the pan, and place over high heat. When the oil is hot enough to sizzle when tested by dipping one end of an eggplant slice into it, slip in as many slices as will fit very loosely, without crowding. Fry the eggplant to a rich golden brown, and use a slotted spatula to transfer to a platter lined with a double thickness of paper towels. Lay the slices flat without overlapping too much. Add more slices to the pan. When they are done transfer them to the platter, after having covered the previous layer with a double thickness of paper towels, and proceed thus until all the eggplants are done. Let cool completely.
Turn on the oven to 400°.
Divide the mozzarella and the pecorino slivers into as many portions as you have eggplant slices. Lay the slices flat on a counter, place on the center of each a portion of mozzarella and pecorino, and sprinkle with pepper. Combine the chopped garlic and parsley and distribute it evenly over the two cheeses. Roll up each slice tightly.
Choose a baking pan that can accommodate all the eggplant rolls in one layer without overlapping. Lightly grease the bottom of the pan with olive oil, and put in the rolls, arranging them so that the folded-over edge of the eggplant faces down.
Wash the cherry tomatoes, cut them in half at their middle, toss them in a bowl with a pinch of salt, then place them over the eggplant rolls, their cut side facing down. Bake for 20 to 25 minutes in the upper third of the preheated oven. Let settle down to lukewarm temperature before serving.