Preheat oven to 375 degrees. Place pie crust in plate on a rimmed baking sheet; bake until crust is golden, about 20 minutes. Cool completely; set aside.
Put a fine-mesh sieve over a medium bowl; set aside. In a medium saucepan, off heat, whisk together 2/3 cup sugar, the cocoa powder, flour, and 1 tablespoon cornstarch. Gradually whisk in evaporated milk. Whisk in egg yolks.
Place pan over medium heat; whisking constantly, cook until the first large bubble forms, about 5 minutes. Reduce heat to low; whisking constantly, cook 1 minute.
Immediately pour mixture through prepared sieve into bowl. Stir in butter and vanilla. Pour warm filling into baked crust; cover surface directly with plastic wrap. Chill at least 2 hours and up to 1 day.
Preheat oven to 450 degrees. Using an electric mixer, beat egg whites, remaining 1/3 cup sugar, and remaining 1 teaspoon cornstarch until stiff peaks form. Spread meringue on top of filling, right up to edge of crust; bake until meringue is lightly browned, 5 to 7 minutes. Serve immediately.