Cook noodles in a large pot of boiling salted water according to package directions. Drain and rinse under cool running water.
Whisk gochujang, soy sauce, brown sugar, tahini, sesame oil, and 2 Tbsp. water in a small bowl to combine; set sauce aside.
Heat vegetable oil in a wok or large nonstick skillet over medium-high. Add broccoli rabe and garlic; season with salt and lots of pepper. Cook, tossing often, until broccoli rabe stems are crisp-tender and leaves are wilted, about 2 minutes. Add reserved sauce and cook, stirring often, until thickened slightly, about 2 minutes. Add noodles and a handful of basil and toss gently until sauce clings to noodles, about 1 minute.
Divide noodles among bowls and top with sesame seeds and more basil. Serve with lime wedges for squeezing over.