Heat the broiler with a rack about 6 inches from the element. In a small bowl, stir and mash the butter and flour to a smooth paste; set aside.
Place the leek pieces on a broiler-safe rimmed baking sheet; it’s fine if some of the layers separate but most should remain intact. Broil, without stirring, until most surfaces are blistered and blackened, about 10 minutes.
Transfer the leek pieces to a blender, then add half of the stock. Blend on high until smooth, about 2 minutes. Strain the mixture through a fine-mesh strainer set over a large saucepan, pressing on the solids to extract as much liquid as possible; discard the solids.
Add the remaining stock to the pan and bring to a boil over medium-high. While whisking, add the butter-flour paste; return to a boil, whisking continuously. Reduce to medium and simmer, whisking occasionally, until the gravy is thick enough to coat the back of a spoon, about 5 minutes.
Add the drippings 1 tablespoon at a time, until flavored to your liking, then season to taste with salt and pepper.
