On a work surface, trim 1/4 inch off each side of 1 eggplant to make a big block, then peel off the remaining skin and halve lengthwise to make 2 thick steaks; repeat with the other eggplant. Lay 1 steak flat and, using a sharp knife, cut a deep, horizontal pocket into the center of one of the long sides, taking care not to cut through the other three sides. Repeat with the remaining steaks. Sprinkle all over with 2 teaspoons salt, then let stand in a colander for 30 minutes.
Meanwhile, in a large saucepan, bring 1 inch water to a boil. Add the spinach and cook until wilted, about 2 minutes. Drain, rinse with cold water, then squeeze out the moisture. Chop, then place in a bowl; stir in the ricotta and 1/4 teaspoon each salt and pepper.
Preheat a grill pan over medium heat. Pat the eggplant dry and brush with 2 tablespoons olive oil; transfer to the grill pan. Cover with foil and cook, turning once, until golden and tender, about 15 minutes. Transfer to a work surface and let cool for 15 minutes.
Meanwhile, toast the ciabatta, then brush the cut sides with the remaining 2 tablespoons olive oil and rub with the halved garlic clove. Cut the bread into 1/2-inch cubes and place in a medium bowl; add the mozzarella and toss.
Position a rack in the upper third of the oven and preheat to 400 degrees . Line a baking sheet with parchment. Stuff one-quarter of the spinach mixture into the pocket of each eggplant steak. Transfer to the baking sheet. Spoon the marinara over the stuffed eggplant steaks. Top with the ciabatta-cheese mixture, mounding it with cupped hands. Bake until the cheese is melted and the bread is golden, about 10 minutes.