Slice the beef as thinly as you can (it's easier if you freeze it for 15 to 30 minutes first). Cut the slices into bite-size pieces.
Heat a large, heavy skillet over high heat until it begins to smoke. immediately add half the oil and all the beef. Cook, stirring occasionally until the beef browns and is of desired doneness, about 3 minutes. Transfer the beef to a bowl and lower the heat to medium-high.
Add the remaining oil to the skillet and add the onion, garlic, and ginger. Stir over medium-high heat until the onion softens and begins to brown, 7 to 10 minutes. Sprinkle with sugar; stir.
Add the beef back to the skillet and stir for 1 minute. Add the soy sauce and lime juice, stir, turn off the heat, and taste, adding more soy sauce if necessary. If the mixture is drier than you like, add a bit of water and heat through. Garnish with the basil and serve immediately.