Bring oil back up to 400 degrees F. Add half the vermicelli to the oil and fry until the noodles start to curl, about 2 or 3 seconds.. Remove the vermicelli from the oil, drain on paper towel and lightly salt. Fry the remaining vermicelli.
Remove the julienned green onion from the ice water and drain on paper towels. Divide the vermicelli equally among 4 serving plates. Divide and arrange the julienned green onions, cucumber, carrot, jicama, and diced tomatoes evenly over vermicelli. Top salads with the pickled onions and the fried taro root.
Sprinkle 1 teaspoon toasted sesame seeds and 1 1/2 teaspoons crushed peanuts over each salad. In a small bowl, combine the edible flower petals, fennel seedlings, daikon sprouts, and fried shallots. Divide the flower-sprout-shallot mixture evenly among salad plates. Serve the salads with the Salted Apricot Dressing alongside.