This recipe is loosely based on the French tarte Tatin -- sliced apples, butter, and sugar on a large pastry crust -- but we kept our apples whole.
Heat oven to 400 degrees. Combine sugar and 1/4 cup water in a saucepan; cook over medium heat until dark amber, 10 to 12 minutes. Swirl pan to prevent crystallization; don't stir. Working quickly, divide hot caramel among 12 mini-muffin cups. Add 1 teaspoon butter to each cup. Set aside.
Combine cranberries and walnuts in a small bowl. Using a teaspoon, fill each apple center with cranberry and walnut mixture. Place apples, stem-side down, into caramel. Cover tightly with aluminum foil, and bake until the apples are tender but still firm, about 20 minutes.
Meanwhile, on a lightly floured surface, roll out puff pastry to a 12-inch square. Using a 2 1/2-inch cookie cutter, cut out twelve circles. Transfer to a baking sheet, and chill until firm, at least 15 minutes. Once apples have cooked 20 minutes, remove foil, and cover each apple with a circle of puff pastry. Return to oven, and continue baking until puff pastry is golden brown and puffed, about 15 minutes. Turn out onto a tray lined with a Silpat (a French nonstick baking mat) or buttered parchment. Garnish with cranberries and walnuts, if desired, and serve.