Fill a large pot with water and add a couple tablespoons of salt. Bring to a rolling boil over high heat. Add linguine and cook according to package directions until al dente (8-10 minutes). Before draining, reserve 1 cup of pasta cooking water. Drain the pasta and set aside.
While pasta cooks, heat vegetable oil in a large skillet or wok over medium-high heat until shimmering. Add ground beef and break it up with a wooden spoon into small pieces. Cook for 5-6 minutes, stirring occasionally, until no pink remains and edges are crispy. Drain off most of the fat, leaving about 1 tablespoon behind.
Add minced garlic to the beef and stir constantly for 30 seconds until fragrant. Watch carefully to prevent burning.
Pour in soy sauce, beef broth, and hoisin sauce. Add brown sugar, ground ginger, black pepper, and red pepper flakes. Stir everything together and let bubble and simmer for 2-3 minutes until sauce reduces and flavors meld.
In a small bowl, whisk together cornstarch and cold water until smooth with no lumps. Pour this slurry into the skillet while stirring constantly. Continue stirring for 1-2 minutes until sauce becomes glossy and thick.
Add drained pasta to the skillet. Using tongs or two forks, toss everything together until every strand is coated with sauce. If sauce is too thick, add reserved pasta water a tablespoon at a time until desired consistency is reached.
Remove from heat and let rest for 2-3 minutes. Top with sliced green onions and sesame seeds. Serve immediately.