In a large heatproof bowl set over a pot of simmering water, melt semisweet chocolate and butter, stirring until smooth. Remove from heat.
Mix pumpkin, milk, brown sugar, eggs, cornstarch, vanilla, salt, cinnamon, ginger, and nutmeg in a medium bowl. Whisk ⅓ pumpkin mixture into chocolate mixture. Whisk in remaining pumpkin mixture until completely incorporated.
Transfer pie dish to a rimmed baking sheet, and pour pumpkin mixture into crust. Bake until center is set but still a bit wobbly, 55 to 60 minutes.
Let cool in pie dish on a wire rack. Refrigerate until well chilled, at least 8 hours (preferably overnight).
Before serving, drizzle melted milk chocolate on top. Serve immediately.Inspired by Martha Stewart