Preheat a grill pan or panini maker to medium heat.
In a large bowl, add red wine vinegar or basmalic, dried oregano, fresh oregano, olive oil, salt and pepper and mix to combine. Add the tomatoes and toss to coat. Set aside.
For the Pesto: in the bowl of a food processor, combine the basil, garlic, and pine nuts. Pulse and chop until almost blended. The resulting pesto will be a little chunky and not too wet. Pulse a few times to chop and combine. With the food processor running, drizzle in the olive oil until combined, The resulting pesto should be a little chunky and not too wet. Add the grated Parmigiano-Reggiano and pulse a couple times just to combine. Season to taste with salt and pepper. Set the sauce aside so that the flavors meld.
To build the sandwiches, lay out a top and bottom piece of ciabatta with the insides facing up. Spread some pesto on both slices of bread. On the bottom piece, add some mozzarella and dressed tomato slices. Top with bread. Brush the outside of the sandwich with a little olive oil. Place on preheated panini maker and close. Cook until golden brown and the cheese has slightly melted, about 3-5 minutes. Continue building sandwiches with remaining ingredients.
Tip: alternatively, cook the panini stovetop in a skillet or grill pan and press with a weighted object.