1. Heat oven to 400 degrees F.
2. Melt butter over medium heat in a large, heavy-bottomed pan. When it foams, add fennel, onions, and carrots. Cook until just soft (2 minutes). Stir in mushrooms and potato. Cook until mushrooms have let off water and have shrunk (6 minutes).
3. Sprinkle flour over vegetables, stir to coat, and cook until raw flavor is gone (1-2 minutes). Stir in broth and milk until mixture is smooth. Bring to a simmer over medium heat, stirring constantly, and cook until slightly thickened (5 minutes).
4. Remove from heat. Stir in peas, herbs, and vinegar. Pour filling into a deep-dish pie pan.
5. Whisk egg together with water. Set aside.
6. Place dough over filling, tuck into the edges of the dish, and remove excess. 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 (25-30 minutes). Let sit to allow sauce to thicken before serving (10-15 minutes).