In a large bowl combine the rice and 2 Tbsp salt. Add water to cover by 1”, set aside. Have ready a lidded pot that measures 10“-11” in diameter and 6“+ deep. Cut 2 rounds of parchment paper the size of the pot.
Season the chicken all over with salt and pepper. In the pot heat 1 Tbsp of olive oil until shimmering. Add the chicken skin side down and cook until browned, about 10 min. Transfer chicken to a plate, cut thighs into large chunks, set aside. Remove the pot from heat. Place 1 parchment paper in bottom and turn to coat with fat.
Add the remaining 3 Tbsp oil to the parchment lined pot, than sprinkle evenly with the almonds. Drain the rice in a fine mesh strainer, then rinse under cool running water and drain again. Scatter 1 cup of rice in a thin even layer over the almonds. In a medium bowl mix together the remaining ingredients, rice through nutmeg. Reserve 1/2 cup of this mixture, then distribute the remainder in an even layer in the pot.
Place the chicken in the pot, slightly nestling the pieces into the rice-cauliflower layer; discard any accumulated juices. Shingle the eggplant slices over the chicken in an even layer. Sprinkle with the reserved 1/2 cup rice.
Pour the broth into the pot (it will not fully cover the eggplant), then bring to a boil. Set the second parchment round over the food, cover the pot with the lid. Cook for 5 minutes, reduce heat to low and cook undisturbed for 35 min.
Remove the pot from the heat, uncover and let stand for 15 min. Remove the parchment paper and then invert a serving platter onto the pot. Holding the platter against the pot, carefully invert the two together; leave the pot overturned on the platter and let rest for about 10 minutes. Slowly lift off the pot and if needed remove and discard the parchment.