Toss the diced eggplant with the salt and set it in a colander for 20 to 30 minutes. Pat thoroughly dry with paper towels.
Preheat the oven as high as it goes, ideally 475 to 500°F (245 to 260°C), with a pizza stone or inverted sheet pan inside for at least 30 minutes.
Heat the olive oil in a skillet over medium-high heat. Add the eggplant and cook for 6 to 8 minutes, until golden and tender.
Stretch the dough into a 12-inch round. Combine the crushed tomatoes with the garlic and spread over the dough, leaving a border for the crust.
Scatter the mozzarella, sautéed eggplant, olives, and capers over the sauce.
Slide the pizza onto the hot stone or pan and bake for 6 to 8 minutes, until the crust is just starting to set and color.
Pull the pizza out, crack the eggs directly onto open spots on the surface, and return it to the oven for 4 to 6 minutes, until the whites are set and the yolks still jiggle slightly.
Finish with torn basil, red pepper flakes, and flaky salt if using. Slice and serve immediately.