Season chicken breasts with salt and pepper, to taste. In heavy skillet, over medium heat, brown the chicken breasts in 2 tablespoons oil, about 3 minutes on each side.
Reduce heat, and add 1/2 cup apple cider. Simmer breasts for about 10 minutes until completely cooked but still soft. Remove from pan, cool, and julienne; reserve.
Heat shallots and cabbage in 2 tablespoons oil, cooking out all moisture. Remove from pan and reserve. Allow any excess moisture to drain.
Add remaining 1/2 cup of cider and pecans to pan and reduce cider to thick glaze. Add apples and toss a for few minutes. Return all ingredients to pan, season with salt and pepper, and cook briefly to remove any free moisture. Set aside; allow to cool before assembling strudels.
To assemble strudels: Brush dough with butter, fold in half. Place about 3 tablespoons of filling on edge of the phyllo; roll into individual strudels, brushing with butter.
Bake strudels at 425 degrees for about 15 minutes, until dough is golden. Allow to rest 3-4 minutes before cutting to serve.