Using the big holes of a box grater, grate the onion. Place it in a strainer set over a bowl. Press firmly with the back of a spoon to squeeze out as much liquid as possible so the meat mixture does not turn watery. Discard the juice and transfer the grated onion to a large mixing bowl.
Add the ground meat, breadcrumbs, garlic, egg, ground cumin, salt, pepper, and parsley to the bowl. Using clean hands, mix just until combined.
Cover the bowl with plastic wrap and refrigerate while you prepare the vegetables.
Trim the stem end of each eggplant, then slice it into ¾-inch-thick rounds. You should end up with 18 to 20 slices total.
Arrange them on a sheet pan and sprinkle both sides with salt. Let them sit until you see beads of moisture on the surface, then dry both sides with paper towels.
Brush each slice with olive oil on both sides.
Preheat the oven to 425 °F (220°C). Grab a 10-inch round (or square) baking dish. The eggplant and patties should fit snugly once they are all standing up, with little room to spare.
Remove the meat mixture from the fridge. Divide it into portions matching the number of eggplant slices you have, so you end up with one patty per slice.
Roll each portion into a ball, then flatten it in your palm into a patty about the same size as your eggplant slices. Keep them as close as you can so everything bakes evenly.
To assemble, starting at the outer edge of the baking dish and working toward the center, stand an eggplant slice up on its edge, then stand a patty next to it.
Continue alternating eggplant and patty in a tight spiral, tucking in green peppers every two or three pairs. As you are assembling, press each piece against its neighbor so they support one another and stay upright. Keep spiraling inward until the dish is full.
Once the spiral is complete, tuck the tomato slices into any remaining gaps around the spiral and place a few in the center (*read note).
To make the sauce, add water, tomato paste, olive oil, garlic, salt, and pepper into a measuring cup and whisk until combined.
Slowly pour the sauce over the arranged vegetables and patties, letting it settle down into the gaps between the vegetables.
Cover the dish tightly with parchment paper, foil, or both layered together. This helps trap steam so the vegetables and meat cook through gently before browning.
Bake in the preheated oven for 45 minutes. Remove the parchment paper, then continue baking for 35-40 minutes more, or until the edges of the eggplant and peppers look slightly charred, the sauce has thickened, and the patties reach an internal temperature of 160°F. The eggplant should be fork-tender throughout.
Let it rest for 5-10 minutes on the counter and garnish with fresh parsley. When serving, spoon the juices from the dish over the eggplant and meat.