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Unfold the two pastry sheets on a large piece of parchment. Cut along one fold line to cut off 1/3 of the pastry sheet. Roll out the larger piece until it is 12 x 6 inches. Roll out the smaller piece lengthwise only until it is 12 inches long, leaving the width as is.
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With a pizza cutter, cut the smaller piece lengthwise strips, each ½ inch wide strips. Two of those strips in half so you will have 4 smaller pieces.
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Place the parchment paper on a large heavy cookie sheet. Brush all of the edges with some of the egg white. Lay two longer strips on the longer edges, then two shorter strips, overlapping the corners of the strips. Repeat with more egg white on top of the strips, they lay two long and two short strips on top of the others. Repeat with the other puff pastry sheet.
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Place the baking sheet with the pastry still on the parchment paper in the refrigerator for at least 30 minutes.
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Preheat oven to 350 degrees. Bake the pastry for 20 minutes. After 10 minutes of baking, prick the bottom of the tarts with a fork very gently to avoid piercing them all the way through. The tart shells should be a very light golden brown.
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Remove the pan from the oven and with a cotton towel, very gently press the center of each tart to flatten it. Remove the towel and cool the crust on the parchment paper on a baking rack for at least 15 minutes. You can prepare the tart 3-4 hours ahead to this point and fill it when you are ready to serve it.
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In a medium bowl, combine mozzarella and sharp cheddar and sprinkle 1/2 cup of the mixed cheeses in each tart. Evenly divide the thyme and sprinkle half of it into the tarts (you will use the other half of the leaves in another layer) along with of all the green onions evenly divided for each tart.
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Cover the tarts with the parmesan and feta and sprinkle with the remaining thyme and olive oil.
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Bake the tarts for 20-25 minutes until they are just golden brown. Remove from the oven and let sit for 5 minutes before cutting each tart lengthwise down the center and then into 8 crosswise strips.
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Each tart will make 16 bite size servings.