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ASSEMBLE THE DESSERT Preheat the oven to 250.
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Butter eight 4-ounce ramekins and sprinkle them with sugar.
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In a large bowl, beat the cream cheese with the sugar until smooth.
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In a medium bowl, beat the eggs with the lemon juice and rum.
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Gradually beat the eggs into the cream cheese.
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Strain into a large measuring cup.
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Pour the lemon mixture into the prepared ramekins.
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Set the ramekins in a baking dish and add enough hot water to the dish to reach halfway up the sides of the ramekins.
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Bake for about 1 hour, or until the flans are set.
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Let cool in the water bath for about 21/2 hours.
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Raise the oven temperature to 325.
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In a heavy, medium, nonreactive saucepan, combine the egg yolks, sugar, and lemon juice.
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Whisk over low heat until slightly thickened, about 5 minutes; do not let boil.
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Remove from the heat and stir in the rum.
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Pour the sauce into a medium bowl, set it in a larger bowl of ice water and let cool, stirring occasionally.
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As the mixture cools, gradually stir in 2 to 3 tablespoons of cold water until the sauce has a pourable consistency.
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Using a lemon zester, strip the zest from the lemons.
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In a small saucepan of boiling water, blanch the zest for 1 minute; drain.
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Repeat the blanching process two more times to remove the bitterness.
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Transfer the zest to paper towels and pat dry.
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In a heavy medium saucepan, combine 1 cup of the sugar with 1/4 cup of water.
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Stir over moderate heat to dissolve the sugar, then boil the syrup until it reaches 238 on a candy thermometer, about 7 minutes.
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Remove from the heat and stir in the lemon zest.
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Drain the zest and transfer to a bowl.
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Toss the zest with the remaining 1/2 cup sugar until coated.
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Line a small baking sheet with parchment paper.
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In a small saucepan, melt the butter over low heat.
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Pour the butter into a small bowl, leaving the milky residue behind.
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Spread 1 sheet of the phyllo dough on a work surface; keep the remaining sheets covered with a damp towel.
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Lightly brush the phyllo with the melted butter.
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Lay another sheet of phyllo on top and brush with butter.
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Repeat the process with the remaining phyllo and butter.
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In a small bowl, combine the sugar and cinnamon.
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Using a 3-inch round pastry cutter or a glass, cut out 8 circles from the phyllo dough and transfer them to the prepared baking sheet.
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Sprinkle the circles with the cinnamon sugar and cover with another sheet of parchment.
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Bake for about 25 minutes until crisp and golden.
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Remove the top layer of parchment and let cool on the baking sheet for up to 2 hours.
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Run a knife around the inside of each ramekin and invert each flan onto a plate.
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Top with a pastry crisp, sugared side up.
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Spoon the lemon sauce around the flans and garnish with the candied zest and mint leaves.