Make the filling: Bring 2 cups milk to a bare simmer in a saucepan over medium heat. Whisk the egg yolks, 3/4 cup sugar and the vanilla in a large bowl until smooth and pale. Whisk in the cornstarch. Slowly whisk the warm milk into the egg mixture, then pour into the saucepan and return to medium heat. Cook, whisking constantly, until the custard thickens and begins to bubble, about 2 minutes. Stir in the butter until melted. Strain the custard through a fine-mesh sieve into a large bowl; set aside to cool, stirring frequently.
Put the remaining 2 tablespoons milk and the chocolate in a microwave-safe bowl; microwave 1 minute. Stir until the chocolate melts, then let cool slightly.
Spread the chocolate over the bottom and up the sides of the crust. Press the bananas into the chocolate, then pour the custard over the bananas. Cover and chill until set, at least 1 hour.
Preheat the oven to 375 degrees F. Make the meringue: Beat the egg whites in a large bowl with a mixer on medium-high speed until foamy. Add the cream of tartar and half of the remaining 1/3 cup sugar and beat until opaque. Add the remaining sugar and beat until stiff peaks form, about 5 more minutes.
