In a large bowl, toss steak with 1 tablespoon sugar, oyster sauce, garlic, and pepper to coat. Let marinate for 20 minutes.
Meanwhile, in a medium bowl, combine lime juice, vinegar, fish sauce, soy sauce, jalapeño, and remaining sugar, stirring to dissolve sugar. Set aside. (Store vinaigrette in refrigerator for up to 1 day.)
In a large wok or skillet over high heat, heat half of oil. Add half of steak to pan and stir-fry until nicely browned, 3 to 4 minutes; remove to plate. Add remaining oil and steak to pan and stir-fry as above. If you are using an electric stove, as opposed to gas, cook the steak in 3 batches, instead of 2, to maintain a high temperature in pan and minimize accumulation of juices. (If excess juice forms, remove steak and leave pan on burner for a short time to evaporate liquid.) Return reserved cooked steak to pan and turn off heat. Add peaches, onion, and basil or mint, and toss to combine.
Divide greens, then the steak mixture, among 8 plates. Drizzle each salad with about 2 tablespoons of reserved vinaigrette.