Put the bulgur in a bowl and add 1/4 teaspoon salt and 1 cup boiling water. Cover and let sit until the bulgur is tender and the water is absorbed, about 30 minutes. Fluff with a fork.
Heat the olive oil in a large skillet over medium-high heat. Lay the eggplant halves cut-side down in the skillet and season with salt, then place another skillet on top and weigh it down with large cans. Cook until the eggplant is soft and golden brown, about 6 minutes. Uncover, flip the eggplant and season with salt, then cover, weigh down and cook about 5 more minutes. Transfer to a paper towel-lined plate to cool.
Pulse the walnuts, onion, celery and garlic in a food processor until coarsely chopped. Add the cilantro, tarragon and coriander and pulse until smooth. Add the bulgur and vinegar and pulse once or twice to combine. Add 1 teaspoon salt, and pepper to taste. Spread the bulgur mixture evenly on the cut sides of 3 eggplant halves; sandwich with the remaining halves. Wrap tightly with plastic wrap and refrigerate at least 1 hour.
Slice the stuffed eggplant into pieces, sprinkle with the feta and drizzle with olive oil.