Remove udon noodles from outer bag. Pierce inner pouch several times and place on a microwave-safe plate. Cook in the microwave until tender, about 90 seconds.
Transfer noodles to a large bowl. Mix in bean sprouts, onion, carrot, mushrooms, cilantro, and sesame seeds.
Mix rice vinegar, soy sauce, olive oil, sesame oil, sugar, ginger, garlic, lime juice, white wine, sriracha sauce, and red pepper flakes together in a small bowl until sugar dissolves. Pour dressing over the noodle mixture in the bowl; toss to combine. Refrigerate until flavors combine, at least 4 hours.
If you are using dried udon noodles, you can cook in a large pot of boiling water, stirring occasionally, until noodles are tender yet firm to the bite, 10 to 12 minutes. Drain and proceed with recipe.