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1
Heat the oven to 400 degrees F and arrange a rack in the middle.
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2
Melt butter over medium heat in a 3- to 4-quart Dutch oven or heavy bottomed saucepan.
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3
When it foams, add fennel, onions, and carrots, and cook until just soft and onions are translucent, about 2 minutes.
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4
Add mushrooms and potato, season well with salt and freshly ground black pepper, and stir to coat.
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5
Cook, stirring rarely, until mushrooms have let off water and are shrunken, about 6 minutes.
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6
Sprinkle flour over vegetables, stir to coat, and cook until raw flavor is gone, about 1 to 2 minutes.
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7
Carefully add broth and milk, stirring constantly until mixture is smooth.
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8
Bring to a simmer over medium heat and cook until slightly thickened, about 5 minutes.
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9
Remove from heat, add peas, herbs, and vinegar, and stir to coat.
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10
Season well with salt and freshly ground black pepper.
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11
Turn filling into an 8 by 8-inch baking dish.
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12
Whisk egg together with 2 teaspoons water and a pinch of salt until evenly mixed.
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13
Set aside.
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14
With kitchen shears, cut dough to fit over the baking dish.
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15
Place dough over filling and tuck into the edges of the dish.
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16
Brush dough with egg wash and cut slits in the top to vent.
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17
Place on a baking sheet and bake until crust is golden brown and mixture is bubbling, about 25 to 30 minutes.
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18
Let sit at least 5 minutes before serving.
