Preheat the oven to 375 F.
Melt the butter in a large pot over medium-high heat, then add the olive oil, garlic and the chicken. Stir them around until the golden, about 3 minutes.
Stir in the carrots and mushrooms and then sprinkle the flour over the top and stir it until it’s all combined with the chicken and vegetables. Cook for 5 minutes, then pour in the chicken broth and wine and stir it around and let it cook and thicken.
Once it starts to thicken add the salt & pepper, and fresh thyme.
Add the cream, then stir the mixture and let it bubble up and thicken, about 3 minutes. Turn off the heat.
Pour the filling into four individual baking dishes (round bowls). Reserve. Let cool.
Divide the dough in four and roll each piece into a circle. Trim the circle to ½-inch or larger than the top of the bowl. Brush the outside edges of each bowl with the egg wash, then place the dough on top. Press the dough so that the edges stick to the outside of the bowl. Use a small cookie cutter or a knife to cut little vents in the surface of the dough.
Bake for 25 to 30 minutes, until the crust is deep golden brown and the filling is bubbly.