Step 1Coat a very large skillet with ¼ cup oil. Place the sprouts in the pan cut sides down, in a single layer if possible, and set over medium heat. Cook without stirring until the bottoms brown, about 5 minutes. Sprinkle generously with salt, stir well, add 2 tablespoons water and cover. Cook covered, stirring once, until the sprouts are tender, about 5 minutes.
Step 2Push the sprouts to one side of the pan and add the remaining tablespoon of oil to the other side. Immediately add the garlic and chile and stir well for 30 seconds, then stir to mix with the sprouts.
Step 3Add the vinegar and cook, stirring, until it evaporates, about 2 minutes. Add the maple syrup and cook, stirring, until the sprouts are glazed, about 1 minute. Serve hot, warm, or room temperature.