Start by rinsing your broccolini under cold water to remove any dirt. Trim off any tough ends from the stems and cut the broccolini into manageable pieces, about 2-3 inches long.
In your wok or large skillet, heat the peanut or vegetable oil over medium-high heat until hot.
Add the minced garlic and ginger to the hot oil. Stir-fry for about 30 seconds until fragrant.
Toss in the prepared broccolini and stir well. Stir-fry for about 2-3 minutes until bright green and tender-crisp.
Pour in the broth, oyster sauce, and chili garlic sauce if using. Stir to combine and cook for an additional 2-3 minutes until flavors meld.
Drizzle the toasted sesame oil over the stir-fried broccolini and give it one last toss.
Remove from heat, transfer to a serving dish, and serve immediately.