Melt the butter in a frying pan. Add the leek and garlic and cook until the leek is softened. Remove the leek from the pan and place in a bowl.
Add the mushrooms and thyme leaves to the pan and cook for a few minutes until gently cooked. Season with salt and pepper to taste.
If making 4 mini tart tatin, I suggest using ramekins. Grease the ramekins with butter. Turn one of the ramekins upside down on the pastry and cut the pastry to size. You will need 8 pastry rounds for 4 ramekins. If making a large tarte tatin you can either use your frying pan if it has a heat proof handle or a spring form pan. If using a spring form pan, grease. Place upside down on the pastry and trim 2 rounds to size. If using your frying pan, trim the pastry to the size of your frying pan.
In whatever container you use, place the cooked mushrooms in the bottom and then layer with the cooked leeks. Top with 2 rounds of pastry and then brush the top layer with beaten egg.
Bake in the oven for 15–20 minutes or until the pastry is puffed and golden. Remove and place a plate or wire rack on top of the container and invert.
Notes
You may also want to try making mini tarte tatin's in a muffin tray for a party appetizer.