Whisk together the hoisin sauce, brown sugar, soy sauce, and applesauce in a small bowl; set aside.
Combine the pork and cornstarch in a bowl. Mix until the cornstarch evenly coats the pork; set aside.
Heat the peanut oil and sesame oil in a large skillet or wok over medium-high heat. Cook the pork in three separate batches in the hot oil until no longer pink in the middle, 2 to 3 minutes per batch. Remove pork to a plate lined with paper towels to drain, reserving the oil. Add the ginger to the skillet; cook and stir for 30 seconds. Stir in the broccoli and cook until tender. Return the pork to the skillet and pour in the sauce; toss to coat. Cook until all ingredients are hot.