Heat the oven to 400 degrees F and arrange a rack in the middle.
butter over medium heat in a 3- to 4-quart Dutch oven
or 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 rarely, until mushrooms have let off water 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, and stir to coat. Season well with salt and freshly ground black pepper. Turn filling into an 8 by 8-inch baking dish.
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. 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 to 30 minutes. Let sit at least 5 minutes before serving.