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Preheat the oven to 375.
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Place the sheet of puff pastry on a parchment-lined sheet tray. Top with a piece of parchment paper, and cover with another sheet tray. This will help keep the pastry from puffing too much.
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Bake the pastry until golden brown and crisp, 20 to 25 minutes. Remove the top sheet tray, and allow the puff pastry to cool.
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While the pastry is baking prepare the topping. To caramelize the onions, heat a sauté pan over medium heat and add the olive oil and butter. Sauté the onions until they are nicely caramelized, stirring occasionally, about 20 minutes. If browned bits form on the bottom of the pan while caramelizing add a splash of water to deglaze.
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Once the onions are fully caramelized to a rich golden color, deglaze with the wine, and season with salt and pepper to taste. Transfer to a medium-size bowl.
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Stir the heavy cream, egg yolks, thyme and cheese into the onion mixture.
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Spread the onion mixture evenly over the bottom of the baked puff pastry.
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Bake until the topping is hot and bubbly, 10 to 15 minutes.
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Transfer the tart to a cutting board, and cut into squares. Top with a sprinkle of parsley, and serve warm or room temperature.
