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To Make the Ice-Cream Base: Combine the milk and cream in a heavy nonreactive medium saucepan.
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Use the back of a knife to scrape the seeds from the vanilla bean and add the seeds and pod to the saucepan.
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Bring to a rolling boil over medium-high heat.
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Let the mixture rise up the side of the pan, then immediately remove from the heat.
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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.
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The mixture will triple in volume and hold its shape when dropped from the whisk.
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It should be very stiff.
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Bring the bowl over to the saucepan.
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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.
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The mixture should be thick enough to coat a wooden spoon.
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(Run your finger through the custard on the back of the spoon.
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The custard is thick enough if the line remains.)
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Pour the custard through a fine-mesh strainer or China cap into a large bowl.
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Place in an ice bath or refrigerate until chilled, stirring occasionally.
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To Make the Banana Ice Cream: Place the ice-cream base, bananas, sour cream, and rum in a large bowl.
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Mix well.
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Add enough sugar to sweeten the mixture, depending on the sweetness of the bananas.
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Add more sugar than seems necessary, as some of the sweetness will be lost during freezing.
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Transfer the mixture to an ice-cream maker and freeze according to manufacturers instructions.
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Preheat the oven to 350F.
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Butter two 8 1/2 x 4 1/2-inch loaf pans and line all sides of the pans with parchment paper.
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To Make the Cake: Sift the flour and baking soda into a bowl and set aside.
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Cream the butter and sugar in the bowl of an electric mixer fitted with the paddle attachment until light and fluffy.
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Add the eggs one at a time and mix well.
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Beat in the vanilla.
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Add the flour mixture alternately with the bananas and sour cream, beginning and ending with the flour mixture, and blending after each addition.
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Pour into the prepared pans and spread evenly.
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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.
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Cool the cakes in their pans on cooling racks for 25 minutes, then invert onto racks and place right side up.
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While the cakes are still warm, brush with the rum.
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The cakes can be stored at room temperature for up to 2 days or frozen for up to 1 month.
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I always sift the dry ingredients through a large sieve set over parchment paper.
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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.
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To Make the Creme 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.
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Let the mixture rise up the side of the pan, then immediately remove from the heat.
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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.
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The mixture will triple in volume and form a ribbon when dropped from the whisk.
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Bring the bowl over to the saucepan.
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Whisk about 1 cup of the hot milk into the yolk mixture to temper it, then pour the mixture back into the saucepan.
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Off heat, stir until the custard is thick enough to coat a wooden spoon.
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(Run your finger through the custard on the back of the spoon.
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The custard is thick enough when the line remains.)
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Immediately pour the creme anglaise through a China cap or fine-mesh strainer into a bowl and whisk until cool.
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Refrigerate for up to 3 days.
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Cover after completely cool.
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To Assemble the Cake: Cut the cake into sixteen 1/2-inch slices.
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Place a cake slice on each plate, top with a scoop of ice cream, and gently press a second cake slice on top.
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Drizzle with the creme anglaise and serve.
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I sometimes like to garnish the plates with either fresh baby bananas, when in season, or sauteed bananas.
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Melt the butter in a skillet until very hot, add the bananas and sugar, and saute until warm and until the sugar has dissolved.
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Serve immediately.
