Preparation

Make the infused milk

• Cut the peeled bananas into 1/2- to 3/4-inch pieces and put them in a medium saucepan along with the milk. Bring to a simmer over medium heat, stirring frequently.
• Slide the pan off the heat, cover, and let the mixture steep for at least 20 minutes or up to 2 hours. (The more bananas used and the longer the steeping, the more pronounced the banana flavor will be.)
• Arrange a fine-mesh sieve over a 4-cup measure and pour the banana-milk mixture into the sieve. Using a silicone spatula, gently press on the banana pieces to squeeze out the milk. Scrape the underside of the sieve, add to the milk, and discard the banana pieces. You should have 2-2/3 cups. If you have less, add more milk and if you have more, pour off some of the milk (save it for your morning cereal). Wipe the saucepan dry—no need to wash.

Make the banana layer

• Cut the peeled bananas lengthwise in half, and then cut crosswise into 1/2- to 3/4-inch pieces. Put the butter in a medium skillet, and cook over medium heat until bubbling. Add the brown sugar and stir until dissolved. Add the banana pieces, and sprinkle with the rum and salt. Increase the heat to medium high and cook, using a silicone spatula to turn and gently stir, until the bananas are soft and the juices are reduced and thick, 5 to 6 minutes. Slide the pan from the heat and set aside, stirring occasionally, until completely cool. (For faster cooling, scrape into a bowl and set the bowl over a larger bowl filled with ice, stirring and scraping the sides frequently until chilled.)

Make the filling

• Scrape the cooled banana mixture into the baked and cooled piecrust and spread evenly.
• Put the sugar, cornstarch, and salt in the saucepan and whisk until well blended. Add 2/3 cup of the infused milk and whisk until the mixture is smooth. Whisk in the egg yolks (or whole eggs, if using), and then the remaining milk.
• Set the pan over medium-high heat and cook, stirring with a whisk, until the mixture comes to a full boil. Reduce the heat to medium and, whisking constantly, boil for 1 minute, then slide the pan off the heat. Add the butter and vanilla, and stir until the butter melts and the pudding is smooth.
• Scrape the filling into the piecrust on top of the banana layer, spread evenly, and press a piece of plastic wrap directly on the surface to prevent a skin from forming, if you like. Refrigerate until chilled and firm, about 5 hours.

Whip the topping

• Put the heavy cream, mascarpone, brown sugar, rum, and salt in the bowl of a stand mixer fitted with the whisk attachment. Beat on low speed until smooth. Increase the speed to medium high, and beat until the cream is thick and holds medium-firm peaks.
• Peel off the plastic from the filling. Scrape the topping onto the center of the pie and, using a small offset or silicone spatula, spread it to cover the filling, leaving the center slightly domed along with lots of swirls and peaks. Arrange banana slices on top of the whipped cream, if you like. Serve immediately or cover loosely and refrigerate for up to 8 hours.