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.

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.

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.

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.

Drizzle with olive oil and sprinkle with parsley before serving.