Bring a large pot of lightly salted water to a boil. Cook spaghetti in the boiling water until cooked through but firm to the bite, about 12 minutes; drain and transfer to a large bowl. Drizzle sesame oil over the spaghetti; toss to coat. Place a plate atop the bowl to keep the noodles warm.
Heat peanut oil in a wok or large skillet over medium-high heat. Cook and stir garlic and ginger in hot oil until fragrant, about 30 seconds. Add mixed vegetables to the skillet; cook and stir until slightly tender, about 3 minutes. Stir flank steak into the vegetable mixture; cook and stir until the beef is cooked through, about 5 minutes.
Mix soy sauce, brown sugar, oyster sauce, and chile paste together in a small bowl; pour over the spaghetti. Dump spaghetti and sauce mixture into the wok with the vegetables and steak; cook and stir until the spaghetti is hot, 2 to 3 minutes.