Preheat broiler. Cut eggplant in half lengthwise. Place the halves cut-side down on a baking sheet. Broil about 4 inches from the heat until the skin is blackened and the flesh is very soft, 10 to 15 minutes. Set aside to cool.
Meanwhile, bring a large pot of lightly salted water to a boil for cooking pasta. Stir sesame seeds in a small skillet over medium-low heat until toasted and fragrant, about 2 minutes. Transfer to a small dish to cool.
Cook noodles until just tender, about 3 minutes. Drain and rinse under cold water until cool. Press to remove excess water, transfer to a large bowl and toss with 1 teaspoon oil to keep them from sticking.
With a chef's knife, mash garlic and salt into a paste. Transfer to a small bowl and add vinegar, soy sauce, ginger, brown sugar, chile-garlic sauce and the remaining 1 tablespoon oil. Whisk until blended.
Peel the cooled eggplant and discard the skin. Chop the eggplant flesh to a coarse puree. Add it to the noodles, along with carrots, cilantro and the sesame seeds. Add the dressing and toss until well combined. Just before serving, garnish with diced cucumber.
Make Ahead Tip: Cover and refrigerate for up to 1 day. Bring to room temperature before serving.
Soba (Japanese buckwheat noodles) add a lovely nutty flavor to the dish, but you can also use linguine.
3 starch, 1/2 other carbohydrate, 1 1/2 vegetable, 1 fat