Using electric mixer, beat butter and sugar in medium bowl until blended. Beat in egg. Add flour and salt and beat just until incorporated. Gather dough into ball; flatten into disk. Wrap in plastic and chill at least 1 hour. DO AHEAD: Can be made 1 day ahead. Keep chilled. Divide dough into 6 equal pieces. Gather each piece into ball, then flatten each slightly. Roll out each dough piece on lightly floured surface to 4- to 41/2-inch round. Press 1 dough round onto bottom and up sides of 41/2-inch-diameter tartlet pan with removable bottom. Pierce crust all over with fork. Repeat with remaining dough rounds. Freeze crusts 30 minutes. Preheat oven to 375°F. Bake crusts until golden brown and baked through, about 30 minutes. Cool crusts in pans on rack. DO AHEAD: Can be made 1 day ahead. Store airtight in pans at room temperature. Combine sugar and 1/4 cup water in heavy medium saucepan. Stir over medium-low heat 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 6 minutes. Remove pan from heat. Add cream, then butter and salt (mixture will bubble vigorously). Stir until any caramel bits dissolve and sauce is smooth. do ahead Can be made 1 day ahead. Cover and chill. Rewarm over medium-low heat until just pourable before using. crumb topping / Blend flour, brown sugar, and salt in processor. Add pecans and blend until coarsely chopped. Add butter and process just until small moist clumps form. DO AHEAD: Can be made 1 day ahead. Cover and chill.Rewarm over medium-low heat until just pourable before using. Blend flour, brown sugar, and salt in processor. Add pecans and blend until coarsely chopped. Add butter and process just until small moist clumps form. DO AHEAD: Can be made 1 day ahead. Cover and chill. Whisk brown sugar and cornstarch in bowl to blend. Melt butter in large nonstick skillet over medium heat. Add apple pieces and saute until almost tender, about 10 minutes. Add brown sugar mixture; stir until sugar dissolves, then add next 3 ingredients. Simmer until juices are bubbling and slightly thickened, about 5 minutes. Remove from heat. Cool completely. Preheat oven to 375°F. Arrange tartlet pans with crusts on rimmed baking sheet. Spoon 11/2 tablespoons caramel sauce into each tartlet crust. Divide cooled apple filling among crusts. Sprinkle crumb topping evenly over each. Bake until apple filling is bubbling and topping is crisp and golden, 25 to 30 minutes (juices may bubble over rims of pans). Cool 5 minutes, then carefully loosen tartlets from sides of pans with small sharp knife and remove pan sides. Cool tartlets 10 minutes, then place 1 warm tartlet on each of 6 plates. Place scoop of cinnamon ice cream alongside each and serve.