For the Dough
To prepare dough, sprinkle yeast over milk in the bowl of a mixer fitted with the paddle attachment. Allow to dissolve for 5 minutes.
Mix remaining ingredients, except butter, on low speed for 5 minutes. Switch to a dough hook attachment and knead on medium speed for 5 minutes.
Add butter a bit at a time until it has all been incorporated. You may have to pull the dough off the hook once or twice to work the butter into it. Knead for 5 minutes more.
The dough may look very soft, but it will come together. Place in a bowl, cover, then chill overnight in the refrigerator.
For the Filling
Heat a large sauté pan over medium high heat and add butter. Once butter halts foaming, add mushrooms and sauté until all liquid has evaporated, about 5 minutes.
Add garlic, oregano and thyme and stir. Add wine or vermouth and cook until absorbed. Remove from heat and stir in red pepper, chives, parsley and season to taste. Let cool, then stir in grated cheese. Filling can be chilled until ready to assemble.
Divide dough into 8 pieces. On a lightly floured surface, roll out 1 portion of dough to just shy of 1/2-inch thick.
Spoon about 1/3 cup of mushroom filling in center and fold over brioche, pinching edges of pastry together. Repeat with remaining dough and place on a parchment-lined baking sheet.
Cover with plastic wrap and let rise for 20 minutes.
Preheat oven to 190°C. Brush parcels with eggwash and bake for 18 to 25 minutes, until pastry is a rich golden brown.