Preheat the oven to 400°F.
Bring a small saucepan of salted water to a boil. Add the carrots and cook for 3 minutes. Drain and set aside.
Melt the butter in a large skillet over medium-high heat. Add the onion and cook until softened, about 3 minutes.
Add the cremini and shiitake mushrooms. Cook until deeply browned, about 5 minutes.
Add the oyster, maitake, beech, and king trumpet mushrooms. Continue cooking until all the mushrooms are tender and richly caramelized, about 5 minutes more.
Stir in the garlic and cook for 30 seconds.
Sprinkle the flour over the mushroom mixture and stir continuously for 1 minute.
Gradually pour in the prepared mushroom broth while stirring. Add the heavy cream, thyme, rosemary, salt, and pepper.
Simmer gently until the sauce thickens but is still slightly looser than gravy, about 4 minutes.
Fold in the carrots, peas, and parsley. Remove from the heat and stir in the sherry vinegar.
Allow the filling to cool for 10 minutes.
Line two individual pie tins with the bottom crusts. Divide the filling evenly between them.
Top with the remaining crusts. Seal the edges and cut several vents in the tops.
Whisk together the egg and water. Brush the crusts with the egg wash and sprinkle lightly with flaky sea salt.
Place the pies on a baking sheet and bake for 30 to 35 minutes, or until the crusts are deep golden brown and the filling is bubbling through the vents.
Allow the pies to rest for 10 minutes before serving.