Bring a pot of salted water to a boil. Add the broccolini and cook until bright green and crisp-tender, 3 to 5 minutes. Remove with tongs to a colander and run briefly under cool water. (Keep the pot of water boiling.) Meanwhile, stir the kimchi brine, sesame oil, brown sugar, garlic and red pepper flakes in a small bowl.
Heat a large nonstick skillet over medium-high heat. Add the vegetable oil and bacon and cook, stirring, until the bacon is crisp, 5 to 7 minutes. Stir in the red onion and cook until browned and softened, about 5 minutes. Stir in the kimchi and cook until softened, about 2 minutes.
Add the noodles to the boiling water and cook until al dente, about 3 minutes. Using tongs, transfer the noodles to the skillet (reserve the cooking water); stir until the noodles are well coated in the bacon-onion mixture.
Add the broccolini and kimchi sauce to the skillet and toss until combined, about 1 minute. (Add up to 1/2 cup of the reserved cooking water if the pasta is too dry.) Season with salt. Divide among plates; top with the toasted sesame seeds and finely chopped kimchi.