Heat the oven to 400 degrees F and arrange a rack in the middle.
Melt butter over medium heat in a 3- to 4-quart heavy bottomed saucepan. When it foams, add fennel, onions, and carrots, and cook until just soft and onions are translucent, about 2 minutes. Add mushrooms and potato, season well with salt and freshly ground black pepper, and stir to coat. Cook, stirring just a little, until mushrooms have shed their moisture and are shrunken, about 6 minutes.
Sprinkle flour over vegetables, stir to coat, and cook until raw flavor is gone, about 1 to 2 minutes. Carefully add broth and milk, stirring constantly until mixture is smooth. Bring to a simmer over medium heat and cook until slightly thickened, about 5 minutes.
Remove from heat, add peas, herbs, and vinegar, red pepper flakes(if using), and garlic powder(if using) and stir to coat. Season well with salt and freshly ground black pepper. Turn filling into an 8"x8" baking dish(I've been told a 9"x13" dish works too).
Whisk egg together with 2 teaspoons water and a pinch of salt until evenly mixed. Set aside.
With kitchen shears, cut dough to fit over the baking dish. Place dough over filling and tuck into the edges of the dish(if you wish, you can make a crust for the bottom too). Brush dough with egg wash and cut slits in the top to vent. Place on a baking sheet and bake until crust is golden brown and mixture is bubbling, about 25-30 minutes. Let sit at least 5 minutes before serving.
To reheat individual servings, warm for 2 min in the microwave, then bake covered for 15 min at 400*F., then remove the cover and bake an additional 5 minutes to crisp up the crust. Otherwise, leftovers will be too soggy or too burnt!