In a mixing bowl, whisk together 1/4 cup soy sauce, 3 teaspoons minced garlic, 3 tablespoons brown sugar, and 1 teaspoon sesame oil to make the sauce.
Heat a large skillet or wok over high heat. Add the 1 pound of cut chicken and pour the sauce over it.
Cook the chicken, stirring occasionally, until no longer pink and it reaches 165°F, about 5 minutes.
In a small bowl, mix 1/4 cup of the 3/4 cup chicken broth with 1 teaspoon cornstarch until smooth.
Add the remaining chicken broth and 3 cups broccoli florets to the skillet with the chicken. Cover and cook until the broccoli is tender, about 3–5 minutes.
Uncover, stir in the cornstarch mixture, and cook over medium-low heat for a few minutes until the sauce thickens.
Stir in 1 cup cashews, toss to combine, and remove from heat. Serve over rice if desired.