Preparation Mix apple cubes and lemon juice in medium bowl. Mix 1 cup sugar, cinnamon, ginger, and white pepper in small bowl to blend. Melt 1/4 cup butter in heavy large skillet over medium-high heat and stir constantly until butter browns, about 3 minutes. Add 1/3 cup water, then sugar mixture and cook until syrup turns deep amber color, stirring constantly, about 4 minutes. Mix in brown sugar and stir until dissolved. Add apple cubes with lemon juice; cook until juices are thick and spoon leaves path when drawn across bottom of pan, about 13 minutes. Add molasses and stir to blend. Divide apple mixture equally between two 14-cube ice-cube trays. Freeze until frozen, at least 4 hours. (Frozen apple cubes can be prepared 1 month ahead. Cover; keep frozen.) Melt remaining 1 cup butter in heavy small saucepan over medium-low heat. Stack 14 phyllo sheets on work surface. Trim phyllo stack to form 13x8-inch rectangle. Cut stack in half lengthwise, forming two 13x4-inch stacks; place on top of each other. Cover with plastic wrap, then damp kitchen towel. Repeat with remaining 14 phyllo sheets, forming second stack. Place 1 phyllo strip on work surface. Brush lightly with melted butter. Sprinkle with 1 teaspoon breadcrumbs. Top with second phyllo strip. Brush lightly with melted butter; sprinkle with 1 teaspoon breadcrumbs. Remove 1 frozen apple cube from ice-cube tray; return remaining cubes to freezer. Place frozen apple cube on lower short end of prepared phyllo strip. Turn bottom right corner of strip over apple cube so that bottom edge lines up with left side. Continue folding up phyllo, flag-style, creating triangle and enclosing apple cube. Brush triangle with melted butter; place on rimmed baking sheet and transfer to freezer. Repeat with remaining pastry strips and apple cubes. (Turnovers can be made 2 weeks ahead. Cover; keep frozen.) Preheat oven to 400°F. Bake strudel turnovers on rimmed baking sheets until golden brown, about 13 minutes. Serve strudel turnovers warm with vanilla ice cream.