Heat another 1 Tbs. of vegetable oil in a wok or extra-large skillet over medium-high. Add the broccoli and cook, stirring occasionally, 5 minutes. Add the remaining 1 Tbs. of oil and cook the chicken until golden brown, 2 to 3 minutes. Push the chicken and broccoli to one side of the pan and add the eggs, using your spoon to quickly scramble them, then stir everything together.