Thaw filo pastry in the fridge overnight. Then let it sit at room temperature for 30 minutes before working, without removing it from its package.
Cook the chicken. Add a splash of water to a small cooking pot, just enough to cover the bottom of the pot. Add the chicken, season with salt, and steam-cook, covered, on low heat for 20-25 minutes. If you have a steam cooker you can use that.
Rest the chicken until it's cool enough to handle.
Preheat oven to 180°C / 350°F.
Heat a splash of olive oil in a frying pan.
Saute onion over moderate heat until completely soft and translucent.
Coarsely dice the mushrooms and add them to the pan. Turn the heat to high and season with salt and pepper.
Cook the mushrooms until they soften and start to sizzle in the pan. Take the pan off the heat.
Stir in the thyme, ground fennel, and nutmeg.
Chop the chicken finely and mix it into the mushrooms.
Combine milk, eggs, and melted butter in a bowl, season with salt and pepper, and set aside.
Lay open the filo stack on a clean working surface and cover it with a tea towel so it won't air-dry while you work.
Brush a 23 x 33 cm (9 x 13-inch) pan with olive oil. Don't use a thick-bottomed baking dish for this recipe because the filo won't bake properly at the bottom.
Add 6 filo sheets to the bottom, crosswise, covering the bottom of the pan and leaving half of each filo sheet hanging out of the pan on each side. Lightly drizzle each filo you add with olive oil.
Spread half of the chicken-and-mushroom filling.
Ladle 3 spoonfuls of the milk mixture on top.
Sprinkle half of each cheese on top.
Add one filo folded in half (it should fit inside the pan this way), sprinkle lightly/brush with olive oil. Repeat with another filo.
Spread the remaining filling on top. Add 3 more spoonfuls (using a ladle) of the milk mixture and sprinkle the remaining cheese on top.
Top with 4 more folded filo sheets, brushing or sprinkling lightly with olive oil on top.
Overlap and fold the bottom filo that hangs out of the pan. Brush with olive oil.
Cut the pie on top and all the way through to the middle (don't cut the bottom filos) and divide it into 10 pieces.
Pour the remaining milk mixture on top.
Bake for approximately 45 minutes on the lower rack of the oven until golden-brown and crispy. If it looks puffy while it bakes, cut the top with a knife to help release steam.
Cool the chicken and mushroom pie for at least 20-30 minutes before cutting and serving.
Cut into 10 large individual pieces or into smaller ones.