Heat 3 tablespoons of avocado oil in a large skillet over medium-low heat. Add the diced onions, and gently cook them until they caramelize, about 45-50 minutes. If your pan gets too dry, add a tablespoon of water to prevent burning. Set aside to cool.

While your onions cook, prepare the eggplant: With a sharp knife or mandolin, slice the eggplants lengthwise into ¼-inch slices. Place them in a colander and sprinkle with 2 tablespoons of salt. Let them sweat in the sink for 30 minutes to remove any bitterness. 

Bring a pot of water to a boil, add the remaining tablespoon of salt, and cook your rice for 8 minutes, or until it is just cooked. Drain and rinse under cold water to stop the cooking process. Set aside.

In a large bowl, combine the mayonnaise, 2 teaspoons of salt, rose water, saffron water, egg yolks, and lemon juice. Add in the rice, and mix well.

Blot the eggplant slices with a paper towel to remove excess salt and any moisture. Heat 1 tablespoon of oil in a large skillet, and fry the eggplant slices until they get a light golden color, and begin to soften, about 2 minutes on each side. Set aside on a paper towel-lined plate and repeat with the remaining slices.

Make the barberry mixture: In a small pot on low heat, melt the butter. Add the sugar and rose water, stirring until the sugar dissolves. Add the barberries, and mix well to coat them with the butter until warmed, 2 minutes. Add the caramelized onions to the mixture and remove from the heat. 

Assemble the tachin: Preheat the oven to 375 degrees, and spray a 7-inch springform pan with cooking spray. Line the bottom with a circle of parchment, and spray again.

Form a circular overlay design with the eggplants along the bottom and up the sides of the pan. Orient the slices so the round end is in the center of the pan, and make sure to overlap the slices so there are no gaps along the corners and up sides. Press down to secure the slices.

Add 2 ½ cups of the rice mixture and smooth it along the bottom of the pan, making sure to pack it down well. Spread 1 cup of barberry mixture in an even layer over the rice. Repeat with the remaining 2 ½ cups of rice mixture, packing down well.

Layer a few eggplant slices on top to secure the rice mixture. Press down so the tachin is densely packed and secure. If there are slices that come up past the sides of the pan, trim them down flush with the pan edge.

Cover the pan with foil, and place onto a baking sheet to catch oil leakage in the oven. Bake for 45-55 minutes. 

Remove from the oven and let the tachin rest uncovered for 10 minutes. Carefully, flip over onto a serving platter, remove the springform pan, and garnish with remaining barberry mixture, slivered pistachios, and rose buds.