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1
In a food processor, grind together the sugar, butter, egg, almonds, extracts and rum until light and fluffy.
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2
Line a shallow 6-inch bowl with plastic wrap and spoon in the nut mixture.
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3
Smooth the top flat and chill in the freezer 30 minutes to form a disk of firm nut filling.
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4
Preheat oven to 475 degrees F.
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5
Meanwhile, roll the puff pastry out a bit thinner just to flatten it and make the sheets square, about 10 inches, then chill them.
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6
Using a pizza cutter and a lid or cake pan as your guide, cut the largest disk possible out of both pieces of pastry.
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7
In a small bowl, whisk together the egg and water to make an egg wash. Place one of the pastry disks on a parchment lined sheet pan and paint a 2-inch border on the edge of the pastry with the egg-wash.
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8
Pop the chilled nut filling out of the bowl, peel off the plastic, and place it in the center of the pastry.
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9
Carefully place the second pastry disk on top and ease it over the nut filling smoothing it out on the edges to match up the bottom piece.
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10
Brush the whole top surface with egg wash.
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11
Using a small sharp knife, score the surface of the pastry decoratively in a spiral pattern.
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12
Bake for 15 minutes, then turn the oven down to 400 degrees F for 30 to 40 minutes more.
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13
Ten minutes before the cake is done, sprinkle it with powdered sugar then finish baking.
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14
Cut into wedges to serve.
