For the vinaigrette, whisk the shallots, garlic, red wine vinegar, balsamic vinegar, mustard, 2 teaspoons salt, and 1 teaspoon pepper together in a small bowl. Slowly whisk in the olive oil, stir in the anchovies (if using), capers, and olives, and set aside.
Core the tomatoes and slice them 1/3 inch thick. Place the bottom halves of the ciabattas, cut side up, on a sheet pan. Place a layer of tomato on each bread, spoon on 2/3 of the vinaigrette, add a layer of mozzarella, and sprinkle with the basil leaves. Spoon the remaining vinaigrette on the cut sides of the ciabatta tops and place them, cut side down, on the sandwiches. Place a second sheet pan on the sandwiches and put weights on top, such as cans of tomatoes. Allow the sandwiches to sit at room temperature for an hour for the flavors to blend.
Heat an electric panini press and toast the sandwiches for 5 to 10 minutes, until the bread is toasted and the mozzarella starts to melt. Cut each sandwich in half and serve warm.