Bring a large pot of salted water to a rolling boil and cook the linguine until al dente according to package directions. Before draining, set aside 2 cups of the pasta water.
In a large skillet set over medium heat, sauté the ground beef along with the minced garlic until browned. If necessary, drain off any excess fat.
Add the ginger, soy sauce, brown sugar, red pepper flakes, hoisin sauce, and black pepper to the cooked meat. Stir well to combine.
In a small bowl, whisk together the cornstarch and cold water until smooth. Pour this mixture into the skillet and let it simmer briefly until the sauce thickens.
Add 1 cup of the reserved pasta water and the cooked pasta to the skillet. Mix thoroughly, allowing the sauce to cling to the noodles. Adjust consistency with additional pasta water if preferred.
Top with green onions and sesame seeds if using. Serve immediately while hot.