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Special equipment: an electric mixer, 3 disposable pastry bags, a 5/8-inch plain pastry tip and a pastry brush
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For the pastry cream: Soak the gelatin sheets in 2 cups of water until soft.
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Combine the egg yolks, sugar and 1/2 cup of the heavy cream in a medium saucepan.
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Bring to a rolling boil, constantly stirring so the yolks do not cook.
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Remove from the heat, add the gelatin and stir until dissolved.
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Let cool to room temperature.
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Add the remaining 1/2 cup heavy cream to the bowl of an electric mixer fitted with a whisk attachment and whip to soft peaks.
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Gently mix into the cooled yolk mixture and chill overnight.
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For the pate a choux: Preheat the oven to 425 degrees F and line 2 large baking sheets with parchment.
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Bring the butter, sugar, salt and 1 cup of water to a boil in a medium saucepan.
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Remove from the heat.
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Quickly stir in the flour with a wooden spoon.
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Return to medium-high heat, stirring constantly, until the mixture pulls away from the sides and a film forms on the bottom of the pan, about 3 minutes.
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Transfer the mixture to the bowl of an electric mixer fitted with a paddle attachment.
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Mix on low speed until slightly cooled, about 1 minute.
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Increase the speed to medium and add the 4 whole eggs, 1 at a time, until a soft peak forms when the batter is touched with your finger.
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If a peak does not form after the 4 eggs, lightly beat the remaining egg white and mix it into the batter a little at a time until it does.
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Transfer the pate a choux to a pastry bag fitted with a 5/8-inch plain tip.
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Pipe the dough into 2-inch rounds about 1 inch apart onto the prepared baking sheets.
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Gently smooth the pointed peaks with a moistened finger, rounding the tops to ensure even rising.
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Bake until puffed and golden brown, about 25 minutes.
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The profiteroles should feel light and airy.
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Transfer them to wire racks to cool completely.
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For the apple jam: Bring the sugar and 3 tablespoons of water to a boil in a large pot over medium-high heat, stirring to dissolve the sugar.
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Continue to cook, swirling the pan occasionally and brushing down the sides of the pan with a wet pastry brush, until the mixture turns a deep amber color, about 4 minutes.
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Add the apples, vanilla bean and scraped seeds, if using, and stir to coat.
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Some of the caramel will seize, but thats okay because its just going to melt down again.
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Reduce the heat to medium and stir occasionally to help the apples cook evenly and dissolve any pieces of caramel.
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Continue to cook until the apples are translucent and softened (some apples will hold their shape, but should still be softened), and most of the liquid has evaporated, 20 to 25 minutes.
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Add the lemon juice and stir to combine.
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If desired, break up any remaining larger pieces of apple with a wooden spoon for a smoother consistency.
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Discard the vanilla bean and store the apple jam in a glass jar or container in the refrigerator for up to one month.
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To serve: Transfer the pastry cream and apple jam to 2 separate pastry bags.
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Snip off the ends and pipe equal parts pastry cream and apple jam into each profiterole.
