Combine light soy sauce, 1/4 cup vegetable oil, garlic, and ginger root in a resealable plastic bag. Add flank steak; seal bag. Marinate in the refrigerator for up to 2 hours.
Preheat an outdoor grill for medium-high heat and lightly oil the grate.
Toss mango, salad greens, bean sprouts, carrot, and snow peas together in a large bowl. Add peanuts, mint, cilantro, Thai basil, and green onion.
Transfer steak to grill, discarding marinade; cook, turning every 3 to 4 minutes, until beginning to firm and is hot and slightly pink in the center, about 10 minutes total. An instant-read thermometer inserted into the center should read 140 degrees F (60 degrees C). Move steak to a dish; cover with aluminum foil. Allow to rest for 5 minutes.
Whisk lime juice, fish sauce, 2 tablespoons vegetable oil, brown sugar, soy sauce, and shallot together in a bowl to create dressing. Stir in chile pepper. Drizzle dressing over mango-greens mixture; divide onto 2 serving plates.
Slice steak thinly against the grain. Top salads with steak.