Using a vegetable peeler, peel eggplant lengthwise in alternating strips. Slice crosswise, 1/2 inch thick. Layer slices in a colander, sprinkling each layer generously with salt. Set colander over a bowl and weight with a heavy object. Transfer to refrigerator, and let stand overnight.
Preheat the oven to 400 degrees. In a large cast iron skillet, heat 1/2 inch oil over high heat. Remove eggplant from colander, and pat dry with paper towels. Working in batches, add eggplant to skillet. Cook until golden brown, turning once. As each batch is finished, transfer to a paper towel-lined baking sheet to drain.
On a nonstick rimmed baking sheet, arrange 3 slices of eggplant in an overlapping circular pattern. Repeat process until you have used half of the eggplant. Sprinkle with a small pinch of oregano. Top each with 1 1/2 tablespoons of each cheese. Using a spray bottle filled with water, lightly spray cheese to moisten. Top with tomato slices, and season with salt. Repeat process using remaining ingredients to make a second layer. Top each with 1 1/2 tablespoons Baita cheese. Spray with a little more water. Add 2 tablespoons water to baking sheet.
Bake until the cheese is bubbling and beginning to brown. Serve immediately.