Roasted Pear Tarte Tatin with Brown Sugar-Balsamic Swirl Ice Cream

Position 1 rack in center and 1 rack in top third of oven and preheat to 375°F. Sprinkle sugar evenly over bottom of heavy 9-inch diameter cake pan with 2-inch-high sides.Scatter butter cubes over sugar, then drizzle with light corn syrup. Arrange pear halves, cut side up and narrow end pointing toward center, snugly in cake pan (pears may not lie flat, but will shrink during cooking and fit evenly).
Place pan on center rack in oven. Bake pears until tender and dark brown in spots, about 2 3/4 hours.
Meanwhile, line large baking sheet with parchment paper. Unfold thawed puff pastry sheet on work surface. Using another 9-inch-diameter cake pan as guide, cut 9-inch round from pastry sheet. Place pastry round on prepared baking sheet. Place baking sheet on upper rack in oven and bake pastry round until puffed and golden brown, about 20 minutes. Cool pastry round completely.
Using slotted spoon, carefully lift pears from syrup in cake pan and transfer to large plate to cool. do ahead Pears and pastry round can be made 4 hours ahead. Reserve cake pan with syrup. Let pears, pastry, and syrup stand at room temperature. Before serving, place pastry round, flat side up, on platter. Carefully arrange pears, cut side down and narrow end in center, atop pastry round. Place pan with syrup over medium-high heat. Boil until syrup turns dark amber color, whisking occasionally, 2 to 3 minutes. Remove from heat. Add pear nectar (mixture will bubble up). Whisk until caramel is smooth, then spoon over pears.
Since this is a special meal, splurge a little by pouring a dessert wine. With the pear tarte Tatin, we like the 2005 Dolce from Napa Valley (California, $85 for 375 ml). This Sauternes-style wine is rich and elegant with dried apricot, honeysuckle, and toasted almond flavors. Because this is a sweet, intensely flavored wine, each guest needs only a small glass.