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For the Cake:
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Preheat the oven to 350 degrees.
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With an electric mixer fitted with a wire whip, beat the eggs and 1 cup of the sugar on medium-high speed in a large mixing bowl until the mixture is pale yellow and thick and has tripled in volume, about 8 minutes.
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Sift the flour, baking powder, and salt together in a medium-size mixing bowl.
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Fold into the egg mixture and blend thoroughly until smooth.
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Add the vanilla and 2 tablespoons of the butter.
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Mix gently.
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Grease a 17 by 12 inch baking pan (or jelly-roll pan) with the remaining 2 teaspoons of the butter.
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Sprinkle with the remaining 2 tablespoons sugar.
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Pour the cake batter into the prepared pan, spreading it evenly.
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Bake until the cake springs back when touched, about 15 minutes.
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Cool for about 2 minutes, then gently flip it out onto a large piece of parchment paper.
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For the Filling:
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In a mixing bowl, combine the egg yolks, cornstarch, and 1 cup of the heavy cream.
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Whisk to blend well.
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Set aside.
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Combine the remaining 1 1/2 cups cream and sugar in a large heavy-bottom saucepan over medium heat.
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Whisk to dissolve the sugar and bring to a gentle boil, about 10 minutes.
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Slowly add the egg yolk mixture, whisking constantly and cook until it thickens, about 5 minutes.
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Be forewarned: the mixture will break.
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Don't be alarmed.
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Pour it into a glass bowl.
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Press a piece of plastic wrap down over the surface of the mixture to prevent a skin from forming.
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Let cool completely at room temperature.
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When cooled pour the mixture into the bowl of an electric mixer fitted with a wire whip.
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Beat at a medium speed to combine the mixture.
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If it will not combine, warm another 1/2 cup heavy cream and slowly add it to the mixture.
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Whip until you have a thick and creamy custard.
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For the Peaches:
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Heat the butter in a large saute pan over medium-high heat.
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Add the light brown sugar and cinnamon, stir until it dissolves.
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Add the peaches and cook, stirring, until the sugar starts to turn dark amber in color, 5 to 6 minutes.
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Add the rum, carefully ignite it, and flambe the peaches.
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Remove from the heat and allow to cool for about 10 minutes.
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Place in the bowl of a food processor, pulse several times or until the peach mixture is spreadable.
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To Assemble:
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Place the parchment-lined cake on a work surface.
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Peel off the cooking paper and roll (lengthwise) and unroll the cake in the paper several times.
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Using a small pallet knife, spread the peach mixture evenly over the cake.
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Spread the pastry cream evenly over the peaches.
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Starting at one end (lengthwise) and using the paper underneath, roll the filled cakes as tightly as possible.
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Wrap the entire jelly roll in plastic wrap and refrigerate for at least 4 hours.
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Remove from the refrigerator and slice diagonally 2 inches thick.
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Place on each serving plate.
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Garnish with whipped cream, mint and powdered sugar.
