Bring a large pot of salted water to a boil. Cook the linguine according to package directions until al dente. Drain and set aside.

Heat a large skillet over medium heat. Add the ground beef and cook, breaking it up with a spoon, until no pink remains and edges begin to brown (about 6–8 minutes). Drain any excess fat.

Reduce heat to medium-low and add the minced garlic to the beef. Cook, stirring, until fragrant—about 30–60 seconds.

Stir in the brown sugar, beef broth, soy sauce, hoisin sauce, ground ginger, ground black pepper, and red pepper flakes if using. Bring the mixture to a gentle simmer.

In a small bowl, whisk the cornstarch into the 2 tablespoons of water to make a smooth slurry. Pour the slurry into the skillet while stirring.

Continue to simmer and stir until the sauce thickens and becomes glossy, about 2–3 minutes. Taste and adjust seasoning—add a splash more soy sauce for salt, or more brown sugar for sweetness.

Add the cooked linguine to the skillet. Toss thoroughly so each strand is coated with the sauce. Warm together for 1–2 minutes so the noodles absorb flavor.

Transfer to plates or a serving bowl. Garnish generously with sliced green onions and serve immediately.