In a medium bowl, combine fish sauce, soy sauce, and brown sugar. Cut round steak lengthwise in half, then cut into ⅛-inch thick slices across the grain. Add beef to the fish-sauce mixture, tossing to coat well; cover and refrigerate beef 30 minutes to marinate.
In a 12-inch skillet or wok, heat 1 tablespoon oil over high heat until very hot. Add beef mixture and cook, stirring frequently (stir-frying), just until the beef is no longer pink, about 1 minute. Transfer beef to a bowl.
Add remaining 1 teaspoon oil to skillet/wok. Add onion and cooking, stirfrying until tender-crisp, about 3 minutes.
Stir in chiles, garlic, and ginger; cook 30 seconds
Return beef to skillet and add basil; heat through.