Start by making the dough. Mix the flour and salt together in a food processor. Add the butter and pulse a few times until the butter is in small pieces. Add the egg and then add the milk gradually and continue pulsing until a lumpy dough forms. If you prefer to make the dough without a food processor, follow the same instructions, but mix all the ingredients in a large bowl with your hands.
Form the dough into a large ball, flatten slightly into the shape of a disk, cover with plastic wrap, and refrigerate until ready to use.
Preheat the oven to 400 degrees.
For the filling, heat a large nonstick skillet over medium heat, add the butter, and saute the onion for about 6 minutes. Then add the mushroom, thyme, and garlic to the onion and saute for another 5-7 minutes. Increase heat to medium high and add the white wine, salt, and heavy cream and heat for about 3 minutes, then set aside.
After the filling is ready, take the dough out of the refrigerator and roll the dough into a thin sheet on a floured surface. Cut out circles shapes (I used a small overturned bowl to trace around with a knife, a large round cookie cutter could also work).
Place the egg in a bowl and lightly beat with a fork. Using a pastry brush, gently brush the outside 1/2 inch of one of the dough circles with the egg wash. Place about a tablespoon of mushroom mixture inside, top with some Asiago cheese, and fold the circle over. Pinch the edges together with a fork, then repeat with the rest of the circles.
Place on a foil lined baking sheet and lightly brush the tops with the egg wash. Bake for 15-20 minutes or until golden brown.