Make crust: In food processor, combine flour, butter, Parmesan and salt. Pulse until mixture resembles coarse crumbs. Add 1/4 to 1/2 cup cold water, 1 tablespoon at a time, until dough comes together and forms a ball. Turn out dough onto lightly floured surface and shape into a disk; wrap in plastic wrap and refrigerate at least 30 minutes or until ready to use. Make filling: In Dutch oven, melt butter on medium-high. Add onion, 1/2 teaspoon salt and 1/8 teaspoon pepper and cook, stirring, 3 minutes. Add carrots, fennel and parsnip and cook, stirring, 3 minutes. Stir in garlic; cook 1 minute. Add flour and cook, stirring continuously, 5 minutes. Add sherry and cook, scraping up any browned bits. When liquid has almost evaporated, stir in broth and then milk. Cook until mixture is thickened and coats back of spoon, about 10 minutes. Remove from heat; stir in chicken, peas and herbs. Season with salt and pepper to taste. Heat oven to 350°F. Grease 6 ramekins or one 9-by 13-inch baking dish with oil. On lightly floured surface, roll dough into 1/4-inch-thick rectangle about 15 inch- by 11 inch. If using individual ramekins, cut 6 circles 1 inch larger in circumference than ramekins. Transfer filling to baking dish(es), lay dough over filling and trim to 1/2 inch from edge of dish(es). Fold overhang under to create a lip and crimp as desired. Cut 1-inch slit in top crust (6 slits for large pie). Beat egg with 1 tablespoon water and brush over crust. Bake until crust is golden brown and filling is bubbling, about 1 hour for large pie, 45 minutes for small pies.