Start by cutting the boneless, skinless chicken breasts into bite-sized pieces. Season generously with kosher salt and pepper to enhance the flavor.
In a large skillet or wok, heat 1 tablespoon of sesame oil over medium-high heat. Once hot, add the chicken pieces in a single layer. Cook for about 5-7 minutes, stirring occasionally, until the chicken is browned and cooked through. Remove the chicken from the pan and set aside.
In the same skillet, add another tablespoon of sesame oil. Add the minced garlic and ginger paste, stirring for about 30 seconds until fragrant.
Add the broccoli florets to the skillet and stir-fry for about 3-4 minutes until they turn bright green and are tender-crisp.
In a bowl, whisk together the chicken broth, brown sugar, soy sauce, oyster sauce, and cornstarch until smooth. Pour this mixture over the broccoli in the skillet.
Return the cooked chicken to the skillet, stirring well to coat everything in the sauce. Let it simmer for about 2-3 minutes until the sauce thickens and everything is heated through.
Drizzle the final teaspoon of sesame oil over the dish for extra flavor. Serve your Chicken and Broccoli hot, either on its own or over a bed of fluffy rice or noodles.
