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Syrup: The morning you are going to bake, combine 4 cups sugar, 2 cups water, 1 cinnamon stick and 2 slices lemon.
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Boil for 15 minutes.
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Remove from heat; remove cinnamon stick and lemon slices, and cool syrup completely. Also may be made a day in advance and refrigerated, covered.
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When syrup is cooled, preheat oven to 350u00b0F
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Brush softened butter inside 9x13x2-inch glass baking dish.
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For the filling, place milk, sugar, semolina, 1/4 stick softened butter, beaten eggs and extract in a heavy 4-quart pot over low heat, and mix.
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Stir continuously until the mixture becomes thicker than pudding consistency.
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Set aside to cool slightly.
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SEE NOTE: To assemble, fold 10 to 12 pieces of phyllo, one at a time, into the buttered baking dish, lightly brushing each piece with melted butter as it is placed. , Make sure to leave half of the sheet lapping over the side of the dish.
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Pour in warm custard filling.
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Layer 10 more phyllo pieces on top, buttering each layer.
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Score the top layer to form roughly 2x2-inch squares. Bake at 350 F for 30 to 45 minutes, or until lightly golden.
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When removed from oven, pour cold syrup over hot Galaktoboureko (sizzling is normal) and allow it to cool to room temperature before serving, 45 minutes.