Individual Carolina Peach Tarte Tatins With Lemon Ice Cream

Combine 1 cup plus 2 tablespoons sugar and 1/4 cup water in heavy medium saucepan. Bring to simmer over medium heat, stirring until sugar dissolves. Increase heat and boil without stirring until syrup is deep amber, occasionally brushing down pan sides with wet pastry brush and swirling pan, about 10 minutes. Remove from heat. Add butter and allow to melt (mixture will bubble vigorously). Stir over low heat until any caramel bits dissolve. Immediately divide caramel evenly among eight 3/4-cup custard cups; cool slightly. Place 1 peach half, cut side up, atop caramel in each custard cup. DO AHEAD:
Preheat oven to 400u0b0F. Place 1 puff pastry sheet on lightly floured surface. Using extra 3/4-cup custard cup as template, place cup upside down on puff pastry sheet. Using small sharp knife, cut around cup, forming dough round. Cut 3 more dough rounds from same puff pastry sheet. Repeat with second puff pastry sheet, making 8 rounds total. Place 1 dough round atop peach in each custard cup, tucking edges of dough around peach in each cup. Brush dough with egg glaze; sprinkle lightly with sugar.
Bake tarte Tatins until pastry is puffed and deep golden, about 25 minutes. Using oven mitts to protect hands, invert 1 tarte Tatin onto each of 8 plates. Let stand until tarte Tatins cool slightly, 10 to 15 minutes. Place scoop of Lemon Ice Cream alongside each and serve.