Thaw the fillo dough completely in its unopened package according to the package directions. Heat the oven to 375°F.

Place one fillo sheet in a 9-inch pie plate, allowing the excess pastry to hang over the edge. Brush it lightly with melted butter and sprinkle with a little granulated sugar.

Add the remaining five sheets one at a time, brushing each with butter and sprinkling with a little sugar. Rotate each sheet slightly so the overhanging edges create an even, layered crust.

Fold or crimp the excess dough around the rim of the pie plate. Bake for about 10–15 minutes, until crisp and golden, then cool the crust completely.

Whisk the granulated sugar, cornstarch, cocoa powder, and salt together in a saucepan. Gradually whisk in the whole milk until the mixture is smooth and free of dry pockets.

Cook the mixture over medium heat, whisking frequently, until it thickens and begins to bubble. Whisk the egg yolks in a separate bowl.

Slowly whisk some of the hot chocolate mixture into the egg yolks to temper them. Return the tempered yolk mixture to the saucepan and cook for about 1–2 minutes more, whisking constantly, until thick and glossy.

Remove the saucepan from the heat and stir in the finely chopped chocolate and vanilla extract until smooth. Strain the custard if desired, then pour it into the cooled fillo crust.

Press plastic wrap directly against the surface of the filling to prevent a skin from forming. Refrigerate for at least 4 hours, preferably overnight.

Whip the cold heavy cream with the confectioners’ sugar and vanilla until soft or medium peaks form. Spread or pipe the topping over the chilled pie just before serving.