Position a rack in the center of the oven and heat the oven to 350°F. On a lightly floured surface, roll out the pastry to a 12x15-inch rectangle.
With a 3-1/2- to 4-inch round cookie cutter, cut out 8 disks of puff pastry. Prick each disk all over with a fork. Arrange them on a rimmed baking sheet lined with a nonstick baking mat or parchment and bake until puffed and golden, about 25 minutes. Set aside to cool. Peel and core the apples with an apple corer. Cut off about 1/2 inch from both ends to create two flat surfaces, and then cut the apple in half along the equator. You should have eight 1/2- to 3/4-inch-thick apple rings. If the rings are wider than 3 inches, use a 3-inch round cookie cutter to trim them down.
Heat 2 Tbs. of the butter in a 12-inch ovenproof skillet over medium heat. Cook 4 of the rings until light golden-brown on the bottom, about 3 minutes. Transfer to a plate. Repeat with the remaining 2 Tbs. butter and 4 apple rings. Flip the apples in the pan and add to them the first batch of apples, browned side up. Bake the apples at 350°F until soft and slightly caramelized, 40 to 45 minutes. Put 1 slice of the cheese on top of each apple and bake until slightly melted, 3 to 4 minutes. Place a disk of puff pastry over the cheese and bake until heated through and the cheese has fully melted, about 3 minutes. Transfer the tarts with a spatula to individual salad plates, turning them over so the apple is on top
In a large bowl, toss the frisée with the walnuts. Whisk the olive oil and balsamic vinegar in a small bowl. Toss the salad with the dressing and season to taste with salt and pepper. Distribute the salad among the plates and serve.