Position racks in the upper and lower thirds of the oven and preheat to 350 degrees F. Line 2 baking sheets with parchment paper.
Melt the butter with the sugar in a saucepan over low heat, stirring, until the mixture thickens and the sugar is almost dissolved, about 5 minutes. Transfer to a large bowl and stir in the oats, flour, vanilla and salt. Let cool 10 minutes, then stir in the egg white.
Drop 6 rounded tablespoonfuls of batter onto each prepared baking sheet, about 3 1/2 inches apart. Spread each mound into a 3-inch circle. Bake, switching the position of the pans halfway through, until golden brown, 12 to 15 minutes. Let cool on the baking sheets until firm, about 3 minutes, then carefully remove using a thin metal spatula and transfer to a rack to cool completely.
Put the chocolate in a small microwave-safe bowl and microwave in 30-second intervals, stirring, until melted. Drizzle on the cookies and let set, about 30 minutes.