Step 1

Preheat the oven to 350°F (175°C).

Step 2

To make the crust, generously butter a 10-inch (25-cm) pie plate. In a medium bowl, mix together the chocolate cookie crumbs, the 3 tablespoons (45 g) sugar, and 4 tablespoons (2 ounces/60 g) melted butter until evenly moistened. Pat the mixture in an even layer into the bottom and up the sides of the buttered pie plate. Refrigerate or freeze for 30 minutes.

Step 3

Bake the crust until if feels slightly firm, about 10 minutes. Let cool completely.

Step 4

To make the filling, in a small saucepan, heat the brown sugar and the 2 tablespoons (1 ounce/30 g) butter, stirring, until the butter is melted. Pour into a large bowl, set a mesh strainer across the top, and set aside.

Step 5

In a small bowl, whisk the cornstarch with just enough of the milk to make a smooth slurry. In a medium saucepan over medium heat, warm the remaining milk and the salt. When the milk is hot, stir the cornstarch mixture to recombine, then whisk it into the milk in the saucepan. Cook over medium heat, stirring constantly, until the mixture begins to boil and thickens. Whisk the egg yolks in a small bowl, then whisk a small amount of the hot thickened milk mixture into the yolks. Whisk the warmed egg-yolk mixture into the saucepan. Cook, stirring constantly, until the mixture returns to a boil and is as thick as mayonnaise. Pour it through the strainer into the butter–brown sugar mixture. Add the 2 teaspoons rum and 1/2 teaspoon vanilla and whisk until combined.

Step 6

Peel and cut the bananas into slices 1/4 inch (6 mm) thick and scatter them in the bottom of the cooled crust. Pour the filling into the crust, cover with plastic wrap, and refrigerate until chilled, about 2 hours.

Step 7

To make the topping, in a stand mixer fitted with the whip attachment (or in a bowl by hand), whisk the cream on high speed until it begins to mound softly. Add the 1 tablespoon sugar and 2 tablespoons rum and continue whisking until the cream holds peaks. Spread or pipe the topping over the pie. Garnish with chocolate curls.

Step 8

The baked or unbaked crust can be refrigerated for up to 4 days, or frozen for up to 1 month. The pie can be refrigerated for up to 2 days, but whip the cream and top the pie with it a few hours before serving.