Special equipment: a broiler-safe 9-by-13-inch baking dish
Position 2 oven racks in the center and upper third of the oven, and preheat to 425 degrees F. Line 2 rimmed baking sheets with parchment.
Cook the oil, tomato paste, garlic and 1/2 teaspoon salt in a large skillet over medium heat, stirring, until the oil turns yellow, about 2 minutes. Add the tomatoes. Rinse the can with about 1 1/2 cups water, and add the tomato water and basil to the skillet. Bring to a simmer, and cook until the sauce thickens, about 20 minutes.
Meanwhile, put the almond flour in a medium bowl. Whisk the eggs and 3 tablespoons water in another medium bowl until frothy.
Lay out the eggplant on a clean surface, and season with 1/2 teaspoon salt and a few grinds of pepper. Dip the eggplant in the egg, press into the almond flour to coat, then place on the prepared baking sheets. Bake until the eggplant is tender and the crust is golden brown, 30 to 35 minutes. Rotate the baking sheets if the top one is browning too quickly.
Turn the oven to broil. Spread a little of the sauce on the bottom of a broiler-safe 9-by-13-inch baking dish. Arrange half the eggplant in the dish in as close to an even layer as you can without overlapping too much. Spread half the remaining sauce on top, then 1/4 cup of the mozzarella. Top with the remaining eggplant, sauce and mozzarella and the Parmesan. Broil until the cheese is melted and browned, about 1 minute. Sprinkle the parsley over the top, and serve immediately, while the cheese is still warm.
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