This dish is flexible: You can eyeball the amounts and variety of veggies, double the recipe, prepare it in advance and freeze it, or make one big casserole. We like individual servings for their elegance—or for kids, so that each can get his or her own special pie. Tie the herbs together with kitchen string or in cheesecloth for easier retrieval, or just pick them out with tongs.
Heat oven to 400º.
Roll out puff-pastry dough and cut to fit tops of 6 8-ounce ramekins or other ovenproof serving pieces. (Be sure to add a ½-inch allowance around each for anchoring.) Lay dough on a parchment-lined baking sheet and refrigerate until ready to use.
In a large saucepan over medium-high heat, melt butter with oil, and saute celery, shallot, and carrots 7-8 minutes. Add stock, bay leaves, thyme, parsley, and chicken. Lower heat and simmer until chicken is fully cooked, about 15 minutes. Remove chicken and set aside to cool.
Raise heat to high; boil until liquid is reduced by half, about 15 minutes.
Chop or shred chicken roughly and return to pan. Add cream and boil 8-10 minutes; then remove bay leaves and thyme and parsley sprigs.
Add green beans, peas, and chopped parsley. Season to taste with salt and pepper.
Divide mixture into ramekins and top with prepared dough. Poke a few holes through the dough to let air escape.
If you like, whisk together egg white and water in a small bowl and brush pastry tops.
Place ramekins on a baking sheet and bake about 20 minutes, or until golden-brown.