Place a pizza stone or baking sheet in oven. Preheat oven to 450°F. Using a vegetable peeler, peel thin strips from the eggplant until you reach the seeds. Discard the seedy core.
Heat 2 tablespoons of the oil in a large nonstick skillet over medium-high. Add the eggplant and cook, stirring constantly, until tender, about 2 minutes.
Place the pizza dough on a sheet of parchment paper. Stretch or roll the dough into a 14-inch circle. Combine the garlic and the remaining 1 tablespoon of oil in a small bowl; brush evenly over the dough. Lay the mozzarella over the dough, leaving a ¾-inch border. Top the mozzarella evenly with the eggplant, chicken, bell pepper, and capers. Sprinkle with the salt. Slide the pizza on the parchment onto the hot pizza stone or baking sheet.
Bake until the mozzarella is melted and the crust is golden brown and crispy, about 15 minutes. Drizzle with the lemon juice and sprinkle with the parsley.