Pour the boiling water into a small bowl and add the beef bouillon. Stir thoroughly until the bouillon has completely dissolved.
Add the soy sauce and brown sugar. Continue stirring until the sugar is fully incorporated, then place the prepared sauce aside.
Slice the beef thinly against the grain. Cut the carrots into 1/4-inch matchsticks, divide the broccoli into approximately 1-inch florets, dice the onion into 1/2-inch pieces, and finely mince the garlic. Keep all ingredients nearby before beginning the stir-fry, since the cooking process proceeds rapidly.
Place a large skillet or wok over high heat and allow it to become very hot, nearly to the smoking point.
Add 2 tablespoons of sesame oil and heat it for approximately 30 seconds, or until the oil begins to shimmer.
Add the thinly sliced beef, working in batches when necessary to prevent overcrowding. Cook for 2–3 minutes, stirring intermittently, until the meat develops browned edges. Allowing the beef to remain undisturbed briefly between turns encourages browning rather than steaming.
Transfer the browned beef to a clean plate and reserve it for later.
Pour an additional tablespoon of sesame oil into the same hot skillet.
Add the diced onion and cook for 2–3 minutes, stirring frequently, until it becomes translucent and lightly golden.
Add the carrot matchsticks and stir-fry for 3–4 minutes, or until they begin to soften.
Add the broccoli florets and continue cooking for another 3 minutes.
Stir in the minced garlic and cook for approximately 30 seconds, stirring continuously until fragrant. Keep this stage brief to prevent the garlic from scorching and developing a bitter flavor.
Return the cooked beef to the skillet with the onion and vegetables.
Pour the prepared sauce over the mixture and gently stir until the beef and vegetables are evenly coated.
Allow everything to simmer together for approximately 2 minutes, giving the sauce time to coat the ingredients and the flavors time to combine.
Add the cooked lo mein noodles to the skillet along with the remaining 1 teaspoon of sesame oil.
Gently toss the mixture for 1–2 minutes, until the noodles are thoroughly heated and evenly covered with the sauce. Avoid excessive stirring so the noodles remain intact rather than becoming overly soft.
Transfer the finished beef lo mein stir-fry to serving bowls or plates while piping hot. Serve immediately while the noodles, beef, and vegetables are freshly heated.