For Pickled Garlic:
Preheat broiler to high (a grill pan also works).
Place Broccolini in a medium bowl; drizzle with oil and juice of ½ a lemon. Sprinkle with salt and black and red peppers, tossing to coat. Add Broccolini to a sheet pan coated with cooking spray. Broil 5 minutes or until charred, turning once. Add lemon slices to the sheet pan and broil another 3 minutes or until charred. Arrange Broccolini and lemons on a platter. Top with additional red pepper, if desired, and pickled garlic.
Bring a small saucepan of water to a boil over high heat. Add garlic and cook for 3 minutes; drain. Transfer the garlic to a 2-cup glass canning jar (or other heatproof jar) with a tight-fitting lid.
Combine water, vinegar, sugar, salt, and spices in a small saucepan. Bring to a boil over high heat, stirring until the sugar and salt are dissolved. Pour the hot pickling solution into the jar. Let cool to room temperature. Cover and refrigerate for at least 8 hours before serving.
Keep refrigerated for up to one month.