Preheat oven to 375 degrees F (190 degrees C). Line a baking sheet with parchment paper.
Bring water and rice to a boil in a saucepan. Reduce heat to medium-low, cover, and simmer until rice is tender, about 20 minutes.
Arrange eggplant slices on the prepared baking sheet. Drizzle olive oil on top; season with salt and pepper.
Bake in the preheated oven until soft, 20 to 30 minutes.
Lay 1 sheet of nori, shiny-side down, on a bamboo sushi mat. Pat half of the rice evenly over nori, leaving edges bare. Arrange 4 slices of avocado and 1/4 cup of the cooked eggplant in a line in the middle of the rice. Wet the edges of the nori with a brush or your fingers. Roll up tightly using the edge of the mat. Repeat with remaining nori, rice, avocado, and eggplant.
Cut each sushi roll in half using a sharp knife. Cut each half into 4 pieces, rinsing off the knife between cuts so you can cut cleanly through the rolls.