Bring a large pot of salted water to a boil. Add the pasta and cook as the label directs. Drain, transfer to a bowl and toss with 1 tablespoon sesame oil.

Mix the soy sauce, sugar, vinegar, garlic, the remaining 2 tablespoons sesame oil and ¼ cup water in a small bowl. Toss 3 tablespoons of the mixture with the beef in a medium bowl. Whisk the cornstarch into the remaining soy sauce mixture.

Heat 2 tablespoons vegetable oil in a large skillet over medium-high heat. Add the broccoli and let brown on the bottom, 1½ minutes, then toss and continue cooking, tossing occasionally, until bright green, 3 to 5 minutes. Season lightly with salt. Remove to a baking sheet.

Add another 1 tablespoon vegetable oil to the skillet and heat until shimmering. Add half the meat and let brown on the bottom, 30 seconds, then turn the meat and cook until the other side browns a bit, another 30 seconds. Add to the baking sheet with the broccoli. Repeat with the remaining 1 tablespoon vegetable oil and meat.

Reduce the heat to medium. Add the water chestnuts and the gochujang to the skillet and toss just to heat, 30 seconds. Stir the soy sauce–cornstarch mixture, add it to the skillet and bring it to a simmer.

Add the broccoli and meat with any accumulated juices to the skillet and toss to coat and heat through, about 1 minute. Serve over the pasta and top with the scallions.