Freeze steak for 15 to 20 minutes, just until the outside is chilled but the steak is not frozen solid. This will make it easier to slice the steak thinly.
Meanwhile, in a medium bowl, combine honey, brown sugar, vinegar, soy sauce, black pepper, and sesame oil. Measure out 1/4 cup of this sauce and pour it into a large zipper-top plastic bag or glass bowl for marinating the meat. Reserve the remaining sauce for the stir-fry.
Remove the steak from the freezer and slice thinly against the grain. Add the steak to the large plastic bag or bowl with the marinade and marinate at room temperature at least 10 minutes or up to 1 hour.
Remove the steak from the plastic bag, discarding the marinade, and pat dry with paper towels. Heat 1 Tablespoon oil in a large skillet over medium-high heat until shimmering. Season half the meat with ¼ tsp salt, add to the skillet in a single layer, and cook for 1 minute without stirring, until browned on one side.
Stir the meat and continue to cook for 1 minute longer, until browned all over. Transfer to a clean bowl. Wipe out the skillet, add another tablespoon of oil, and repeat with the remaining steak.
When the meat has been cooked and set aside, wipe out the skillet. Add the last tablespoon of oil and heat until shimmering. Add the peppers and onions and stir-fry until tender-crisp and still bright, about 3 to 5 minutes.
Meanwhile, stir the cornstarch into the reserved stir-fry sauce. Clear a space in the middle of the vegetables. Add the garlic and ginger to the skillet and mash together, then mix in to the vegetables.
Return the steak and any accumulated juices to the skillet. Pour in the reserved stir-fry sauce and cook over medium-high heat until the sauce boils and thickens slightly, about 2 minutes. Remove from heat and serve over mashed potatoes.