Heat a dual-contact electric grill, a grill pan, or an outdoor grill to medium-high heat.
Meanwhile, on a large platter, drizzle the broccolini with olive oil and sprinkle it with kosher salt, black pepper, and garlic powder. Use your hands to toss everything together until the broccolini is well-coated.
Grill the broccolini, turning it often, until the stalks are tender-crisp and the tops are browned and caramelized. How long you'll need to grill depends on the thickness of the stems. Medium ones should take about 8 minutes. For thin ones, 3-4 minutes is all it takes. Very thick stems can take up to 15 minutes to soften. Lower the heat to medium and cover (if your grill has a lid) to help the stems soften if they are very thick.
Serve immediately.
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Notes
You don't want your grill too hot because then the broccolini will burn before it has a chance to become tender.
Seasonings, especially salt, are guidelines. Adjust to taste.
I prefer garlic powder (or garlic granules) to minced fresh garlic in this recipe. The powder coats the broccolini more evenly and doesn't burn as easily as fresh garlic.
The entire vegetable is edible. You can eat the stems, florets, and leaves. However, for thicker-stem broccolini, trim off the very bottom of the stems if they're tough. I do remove the leaves, as they can burn on a hot grill.
You can keep the leftovers in an airtight container in the fridge for 3-4 days. They become soggy when refrigerated. One way to mitigate that is to place them on paper towels and replace the towels daily. You can serve them cold in a salad or reheat them uncovered in a 350°F oven. I don't recommend reheating them in the microwave because of the sogginess issue.