Pat the sliced beef dry with paper towels, then toss with 1 teaspoon kosher salt and 1/2 teaspoon ground black pepper.
Heat 1 tablespoon sesame oil in a large skillet or wok over high heat until the oil is very hot and shimmering. Add the beef in a single layer and stir or toss constantly until browned and cooked through, about 3–5 minutes. Remove the beef to a plate, cover or tent to keep warm.
Pour off any excess liquid from the skillet, leaving any browned bits. Add the remaining 1 tablespoon sesame oil and return the skillet to high heat until the oil is very hot and shimmering.
Add the sliced yellow onion, sliced red bell pepper, sliced yellow squash, and sliced shiitake mushrooms to the skillet. Stir-fry, tossing constantly, until the vegetables are crisp-tender, about 3–5 minutes.
Add the minced ginger and minced garlic to the vegetables and stir continuously for about 30–45 seconds, until fragrant.
Return the cooked beef to the skillet and stir to combine, cooking just until the beef is reheated, about 1–2 minutes.
Meanwhile, in a small saucepan combine the 1/4 cup BBQ sauce (no sugar added), 2 tablespoons honey, and 2 tablespoons low-sodium soy sauce. Bring to a boil, then reduce heat and simmer for 1 minute.
Pour the sauce over the beef and vegetables in the skillet. Stir constantly until everything is coated and the sauce is heated through and slightly thickened, about 1–2 minutes.
Remove from heat and serve immediately with brown rice or quinoa.