Fill a large pot with water and add 2 teaspoons of the salt. Bring to a boil over high heat. Add the broccoli and cook until bright green and slightly tender, about 2 minutes. Using a slotted spoon, transfer the broccoli to a plate, reserving the boiling water in the pot.
Season the chicken all over with the remaining 1 teaspoon of salt. Add the chicken to a medium bowl along with 2 tablespoons of the cornstarch; toss to evenly coat. To the boiling water, add the coated chicken and cook until just opaque, about 4 minutes. Drain the chicken and set aside.
In a small bowl, combine the soy sauce, sesame oil, garlic, ginger, brown sugar, red pepper flakes, and the remaining 1 tablespoon of cornstarch.
Heat the oil in a large skillet over medium heat. Once the oil is glistening, add the broccoli, onion, bell pepper, and carrots and cook, stirring occasionally, until beginning to brown, about 4 minutes. Add the chicken and sauce and cook, stirring, until the sauce is slightly thickened, about 2 minutes.
Serve topped with green onions and sesame seeds if using.
