To Make the Ice-Cream Base: Combine the milk and cream in a heavy nonreactive medium saucepan. Use the back of a knife to scrape the seeds from the vanilla bean and add the seeds and pod to the saucepan. Bring to a rolling boil over medium-high heat. Let the mixture rise up the side of the pan, then immediately remove from the heat.

Meanwhile, beat the egg yolks with the sugar on high speed in the bowl of an electric mixer fitted with the whisk attachment until very thick and pale, about 5 minutes. The mixture will triple in volume and hold its shape when dropped from the whisk. It should be very stiff.

Bring the bowl over to the saucepan. Whisk about 1 cup of the hot cream mixture into the yolk mixture to temper it, then pour the mixture back into the saucepan and whisk until thoroughly combined. The mixture should be thick enough to coat a wooden spoon. (Run your finger through the custard on the back of the spoon. The custard is thick enough if the line remains.) Pour the custard through a fine-mesh strainer or China cap into a large bowl. Place in an ice bath or refrigerate until chilled, stirring occasionally.

To Make the Banana Ice Cream: Place the ice-cream base, bananas, sour cream, and rum in a large bowl. Mix well. Add enough sugar to sweeten the mixture, depending on the sweetness of the bananas. Add more sugar than seems necessary, as some of the sweetness will be lost during freezing. Transfer the mixture to an ice-cream maker and freeze according to manufacturer’s instructions.

Preheat the oven to 350°F. Butter two 8½ × 4½-inch loaf pans and line all sides of the pans with parchment paper.

To Make the Cake: Sift the flour and baking soda into a bowl and set aside.

Cream the butter and sugar in the bowl of an electric mixer fitted with the paddle attachment until light and fluffy. Add the eggs one at a time and mix well. Beat in the vanilla. Add the flour mixture alternately with the bananas and sour cream, beginning and ending with the flour mixture, and blending after each addition.

Pour into the prepared pans and spread evenly. Bake for 20 minutes, rotate the pans from front to back, and bake for another 5 minutes, or until the cakes pull away slightly from the sides of their pans, are firm to the touch, and are deep brown in color. Cool the cakes in their pans on cooling racks for 25 minutes, then invert onto racks and place right side up. While the cakes are still warm, brush with the rum. The cakes can be stored at room temperature for up to 2 days or frozen for up to 1 month.

I always sift the dry ingredients through a large sieve set over parchment paper. Then when I am ready to add the flour to my batter, I lift the two sides of the paper and hold them together to form a cradle and pour it in.

To Make the Crème Anglaise: Place the milk in a heavy nonreactive medium saucepan, scrape the half vanilla bean seeds in, and bring to a boil over medium-high heat. Let the mixture rise up the side of the pan, then immediately remove from the heat.

Meanwhile, in the bowl of an electric mixer fitted with the whisk attachment, beat the egg yolks with the sugar on high speed until very thick and pale, about 3 to 4 minutes. The mixture will triple in volume and form a ribbon when dropped from the whisk.

Bring the bowl over to the saucepan. Whisk about 1 cup of the hot milk into the yolk mixture to temper it, then pour the mixture back into the saucepan. Off heat, stir until the custard is thick enough to coat a wooden spoon. (Run your finger through the custard on the back of the spoon. The custard is thick enough when the line remains.) Immediately pour the crème anglaise through a China cap or fine-mesh strainer into a bowl and whisk until cool. Refrigerate for up to 3 days. Cover after completely cool.

To Assemble the Cake: Cut the cake into sixteen ½-inch slices. Place a cake slice on each plate, top with a scoop of ice cream, and gently press a second cake slice on top. Drizzle with the crème anglaise and serve.

I sometimes like to garnish the plates with either fresh baby bananas, when in season, or sautéed bananas. Melt the butter in a skillet until very hot, add the bananas and sugar, and sauté until warm and until the sugar has dissolved. Serve immediately.