Bring a large pot of salted water to a boil over high heat. Add the noodles and stir to keep them from sticking together. Cook until barely tender and still firm, 3 to 4 minutes. Drain immediately and rinse under cold running water to cool. Drain the noodles really well, transfer to a wide serving bowl, and toss immediately with 3 tablespoons of the sesame oil so they don't stick together.
Meanwhile, make the sauce. In a saucepan, heat 1/4 cup sesame oil over medium-low heat. Add the ginger, garlic, and chile. Cook, stirring for a minute, until the vegetables are soft and fragrant. Place into a blender along with the brown sugar, peanut butter, vinegar, soy sauce, hot water, chili sauce, and the remaining 2 tablespoons sesame oil. Puree and refrigerate until cold.
Toast the sesame seeds in a dry skillet over medium-low heat until golden brown. (Shake the pan constantly to keep the seeds from burning). Put the noodles in a serving bowl and toss with the sesame sauce. Sprinkle with the sesame seeds and green onions, and serve with the lime wedges.