Fill a large bowl with ice water. Fill a large pot with water and bring to a boil over high heat. Add broccolini to a pot and cook until crisp tender and still bright green, about three minutes. Transfer broccolini to ice water to chill. Once broccolini is cool, transfer to a colander and drain.
Add garlic, chiles, fish sauce, sugar, and sesame oil to a large bowl and stir until thoroughly combined and sugar has dissolved. Combine broccolini and fish sauce mix in a large resealable food storage bag or vacuum food bag and seal. Refrigerate immediately.
Build a two-zone fire in a charcoal or gas grill or fire ring. For a gas grill set half the burners to high medium-high heat and the other half to low. For a charcoal grill or fire ring, bank the hottest coals up and to one side of the cooking area.
Place broccolini on the hot side of the grill.