In a medium bowl, toss the chicken pieces with the cornstarch and ½ tsp of the salt.
Heat 3 tablespoons of oil in a large skillet or wok over medium-high heat. Once the oil is shimmering hot, add the chicken in a single layer. Cook it on the first side until it is golden, about 5 minutes, then flip and push the chicken to one side.
Add the garlic and ginger to the empty side of the pan and let it cook for about 30 seconds, until fragrant.
Stir the remaining 1 tbsp cornstarch into the chicken broth.
To the pan with the chicken, add the broccoli, remaining salt, chicken broth, soy sauce, oyster sauce, sesame oil and black pepper. Give it a good stir and cover.
Cook until the broccoli turns bright green, about 3 minutes, then uncover and continue cooking until it is crisp-tender and chicken is cooked through to 165ºF, about 2 minutes more.
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Notes
Leave the chicken chunks large (like 2-bite pieces; or 1 ½ oz per cube). That way, the chicken won’t dry out by the time the broccoli is finished cooking.If you like a sweeter stir fry sauce, add 1 tsp honey with the other ingredients.