Prepare the filling: In a medium saucepan, melt the butter over medium heat. Add the sugar, flour, and cocoa powder, stirring for 2 minutes until smooth.

Add the milk: Gradually whisk in the milk and bring to a simmer, stirring constantly. Cook for 5 minutes until the mixture thickens.

Temper the egg yolks: Whisk the egg yolks in a separate bowl, then slowly add a small amount of the hot chocolate mixture to the yolks. Gradually pour the tempered yolks back into the saucepan, stirring constantly.

Finish the filling: Cook for another 2-3 minutes, until it thickens to a pudding-like consistency. Remove from heat and stir in vanilla. Pour the filling into the baked pie crust.

Prepare the meringue: Beat the egg whites and cream of tartar at medium speed until soft peaks form. Gradually add the sugar, one tablespoon at a time, until stiff peaks form. Stir in vanilla extract.

Top the pie: Spread the meringue over the chocolate filling, making sure to seal the edges. Use a spoon to create peaks and swirls on top.

Bake the pie: Preheat the oven to 350°F (175°C) and bake the pie for 12-15 minutes, until the meringue is golden brown.

Cool and serve: Let the pie cool to room temperature for at least 1 hour before serving.