Make the dough: Mix the flour and salt in a bowl, cut in the butter by hand or by pastry cutter until small balls form.
Add the egg and water to flour and mix until a dough forms.
Roll the dough out into a thin sheet and cut rounded discs (3") until all the dough is used.
Make the filling: Melt the butter over medium heat and to it add the mushrooms, carrots, and shallots cooking for 15 minutes or until tender.
Add balsamic vinegar and cook about 3 minutes then allow the mixture to cool completely.
Assemble the pies: place a spoonful of the mushroom mixture and a spoonful of grated cheese at the center of the dough disc.
Brush the edge of the dough with egg then fold and crimp with fingers or a fork. Repeat until all filling and dough is used.
Brush the tops of each pie with egg.
Bake in a 200°C oven for 20 minutes until golden.