Preheat oven to 400 degrees F.
Heat 2 tbsp of butter in the cast iron skillet. Add 1/2 the mushrooms. Sprinkle with salt and cook until slightly browned. Add wine and cook until the wine is completely absorbed. Remove from pan and set aside. Repeat the process for the second batch of mushrooms and also set aside.
Heat second 2 tbsps of butter in the skillet. Add the leeks, season with salt, and cook until tender, about 3 minutes. Add in potatoes and the mushrooms. Sprinkle mixture with flour and stir until vegetables are evenly coated. Add pheasant, thyme, chicken broth, and cream. Bring to a simmer and cook for 15 - 20 minutes over low heat.
While the filling simmers, roll out the pastry and cut it into a circle (or circles) big enough to cover your pan (or pans).
Remove the skillet from heat. Top with puff pastry. Brush with egg wash. Bake for 15 - 20 minutes or until the puff pastry is golden brown and puffed up, and the filling is hot and bubbly.
Carefully remove the skillet from the oven. Allow to cool for 5-10 minutes before serving. Garnish with fresh thyme.
