Cut the steak into thin slices across the grain, then cut into 2-inch lengths. Toss with the cornstarch and 3/4 teaspoon salt in a medium bowl. Heat 1 tablespoon of the oil in a large nonstick skillet over medium-high heat. Stir-fry the steak, in 2 batches, until well browned (but not cooked through), 4 to 5 minutes per batch, adding 1 more tablespoon of the oil between batches. Transfer the steak to a plate; reserve the skillet.
Reduce heat to medium. Add the remaining tablespoon of oil to the skillet. Add the bok choy. Cook, stirring occasionally, until crisp-tender, 3 to 4 minutes. Add the snap peas, scallion whites, and chili. Stir-fry until fragrant, about 1 minute.
Add the steak, soy sauce, brown sugar, and 3 tablespoons water to the skillet. Cook, tossing frequently, until everything is coated with the sauce, 2 to 3 minutes.
Serve over the rice, topped with the scallion greens and basil.
Excellent. A keeper.