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