To make the dressing: Whisk together the shallot, vinegars, sugar, water, sesame oil, lime juice and salt. Taste it and season with salt if you find it necessary. Set it aside.
For the Salad: Steam the asparagus until it is bright green and still crisp, about 5 minutes. Let it cool for 10 minutes and slice them into 2 inch pieces.
Sautee mushrooms in a skillet until the moisture released by the mushrooms has evaporated and they are golden brown, 7 to 8 minutes.
Place the mushrooms and asparagus into a serving platter and evenly drizzle it with the dressing.
When ready to eat sprinkle it with toasted sesame seeds.