On a lightly floured surface, roll the pastry to an 11-inch square; cut into 4 equal squares. Using a paring knife, lightly score each square (do not cut all the way through), forming a 1-inch border. Using a fork, prick the dough inside the border.
Place the pastry squares on a baking sheet; refrigerate 10 minutes. Then bake until golden, about 15 minutes.
Meanwhile, in a large skillet, heat the oil over medium-high. Add the mushrooms; season with salt and pepper. Cook, tossing frequently, until tender, 6 to 8 minutes.
Dividing evenly, sprinkle ½ cup of the Romano cheese in the center of the baked pastry squares; top with mushrooms and remaining ¼ cup of cheese. Bake until the cheese is melted and lightly browned, 6 to 8 minutes. Sprinkle with chives, and serve.