Heat 1 tablespoon of oil in a heavy-bottomed wok or large skillet over high heat. Once hot, add the minced garlic and stir-fry for 15 seconds until fragrant but not browned.
Toss in the sliced chicken and stir-fry until the pieces lose most of their pink color and start turning white. Add the Chinese broccoli stems and continue stir-frying until the chicken is nearly cooked through and the stems begin to soften.
Add the broccoli leaves and stir-fry briefly until just wilted. Push all the contents to one side of the wok. Crack the egg directly into the empty side and scramble it quickly until just set.
Remove all the contents of the wok onto a plate and scrape the wok clean. Return the wok to the heat, increase to high, and add the remaining 2 tablespoons of oil. Wait until the oil begins to smoke—this high heat is essential for achieving that signature wok char.
Add the noodles along with the sauce and toss minimally—just enough to coat the noodles and let their edges sear and caramelize. This should take about 1 to 1 1/2 minutes.
Finally, return the chicken and vegetable mixture to the wok and toss everything together briefly to combine. Serve hot for best texture and flavor.