Bring a pot of water to a boil. Meanwhile, whisk the egg white, mirin and 1 tablespoon cornstarch in a medium bowl. Add the chicken and toss to coat. Whisk the remaining 2 tablespoons cornstarch, the oyster sauce and 1/2 cup water in a small bowl.
Add the broccolini to the boiling water and cook until crisp-tender, 1 to 2 minutes. Drain and rinse under cold water.
Heat 1 tablespoon olive oil in a large nonstick skillet over medium heat. Add the chicken and cook, stirring, until no longer pink, about 4 minutes. Transfer to a plate. Increase the heat to high and add the remaining 1 tablespoon olive oil. Add the garlic and ginger; stir-fry 30 seconds. Add the scallions and peppers; stir-fry 1 more minute. Add the chicken, broccolini and nuts and stir-fry 2 more minutes.
Reduce the heat to medium; whisk the oyster sauce mixture and add to the skillet. Cook, stirring, until thickened, about 3 minutes.