Butter four 8 oz. ovenproof ramekins and set aside.
Combine chicken breast halves with vegetable broth in medium Dutch oven over high heat. Bring to boil, reduce heat and simmer for 15 minutes. Remove from heat, remove chicken fromm pot, and strain liquid to remove solids. Sit chicken and borth aside.
Heat oil in Dutch oven over medium high heat. Add onion, carrots, celery, and potatoes, and saute, stirring frequently, until beginning to brown. Add mushrooms, allow them to release their liquid but not evaporate. Whisk in flour and cook for one minute.
Whisk in reserved broth. Add rosemary and thyme leaves. Bring to boil, then reduce heat and allow to simmer and thicken for 10 minutes.
Meanwhile, shred cooled chicken and return to pot. Stir and season to taste with salt and freshly ground pepper. Cover and simmer over low heat to combine flavors. If liquid seem too thin, remove lid and reduce until thickened. If liquid seems to thick, add 1/4-1/2 cup of additional vegetable broth and/or dry white wine,.
Place 3 sheets of phyllo on work space, cover remaining 3 sheets with plastic wrap, then a moist towel. Carefully brush each layer of phyllo with melted butter. Sprinkle the top layer with salt and freshly ground black pepper. Carefully cut phyllo stack in half crosswise and stack to make 6 layers. Cut 2 pieces to fit the inside top of each ramekin. You can place a ramekin upside down on the dough to use as an outline to trim the dough. Repeat with remaining phyllo dough.
Spoon chicken mixture into buttered ramekins. Carefully place prepared phyllo on top of chicken so that phyllo edges are inside ramekins.
Bake in oven for 15-20 minutes, until bubbly and phyllo is browned.