Prepare the loaf pan by lining the bottom and sides with parchment paper. Butter the ends of the pan to keep the loaf from sticking.
Preheat your grill to high heat or the broiler on your oven. Pierce the eggplant a few times with a fork or knife. Place the eggplant on the grill or under the broiler. If using the grill, put the cover down. Cook the eggplant for 25-30 minutes, turning every 8-10 minutes, until completely charred on all sides and the flesh is soft when pressed. Transfer it to a platter and cool for about 10 minutes.
Preheat oven to 325℉.
When the eggplant is cool enough to handle, scrape the flesh of the eggplant out into a large bowl. Add the eggs, oil, and vanilla extract to the bowl and mix thoroughly with the eggplant.
In a medium bowl, whisk together the flour, brown sugar, granulated sugar, baking powder, baking soda, and salt. Add the flour mixture to the eggplant mixture and mix until combined. Add the pistachios and stir together.
Transfer the batter to a prepared loaf pan (8½-inch-by-4½-inch). Bake in a preheated oven at 325℉ for 1 hour to 1 hour 20 minutes, or until a toothpick inserted into the center comes out with just a few crumbs sticking to it.
Let the bread cool for 10 minutes on a wire rack. Flip the pan over to remove the bread and let cool completely on a wire rack for 40 to 50 minutes.