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Heat oven to 360 degrees F (180 C). On a lightly floured surface, roll puff pastry into a 10-by-14-inch rectangle. Using a paring knife, lightly score a border around the perimeter of the puff pastry about a ¼-inch away from the edges. Place the puff pastry on a parchment-lined baking sheet and prick the pastry inside the border using a fork to prevent puffing in the center. Bake on the top rack until puff pastry is lightly golden, about 20 minutes. Remove from the oven and let cool slightly.
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Meanwhile, toss carrots with 1 tablespoon oil, season generously with oregano, salt, and pepper, and spread into a single layer on a baking sheet. Roast carrots on the bottom rack (underneath the puff pastry) until the edges are golden brown and carrots are still crisp-tender, 15 to 20 minutes.
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While puff pastry and carrots are in the oven, blend ricotta, manouri, and garlic in a food processor until smooth. Season with salt and pepper.
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Spread the cheese mixture onto the puff pastry up to the border and arrange the carrots in a single layer on top. Bake until the carrots are tender and the edges of the cheese mixture are golden brown, 15 to 20 minutes.
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Drizzle with olive oil and sprinkle with parsley before serving.