Melt the butter in a medium skillet. Add the apples and cook over moderate heat, stirring, until softened and browned, about 5 minutes. Add the sugar, cinnamon and 1/4 cup of water and cook for 2 minutes longer. Stir in the dulce de leche until melted. Scrape the mixture into a bowl and refrigerate until chilled.
Fold the apple mixture into softened vanilla ice cream and freeze until firm, about 4 hours. Scoop into bowls and garnish with chocolate shavings and crumbled gingersnaps.