Heat oven to 375 degrees F (190 C). Line a rectangular 9-in X 13-inch baking pan or tart pan with parchment paper and set aside

Toss carrots with 2 tablespoons of olive oil, oregano, garlic paste, salt, and pepper, and spread into a single layer on a baking sheet. Roast carrots on the bottom rack until the edges are golden brown and carrots are still crisp-tender, 15 to 20 minutes, tossing occasionally to keep them from charring. Remove the carrots and increase the oven temperature to 450F (220C).

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. Place another baking pan over the top, to weigh down the pastry, and add the beans to the top pan as weight. Bake until puff pastry is lightly golden, about 12 to 15 minutes. Remove from the oven, remove the top pan and weights, and let the pastry cool slightly. Lower the oven temperature again, to 350F (160C).

In the bowl of a food processor, blend the cheeses, garlic and basil until smooth. Add a tablespoon or two of olive oil if necessary to make the mixture creamy. Season to taste with salt and pepper.

Spread the cheese mixture onto the cooled 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 a tablespoon or two more olive oil and sprinkle with parsley or chives before serving.