Rinse the mushrooms well - they’re firm enough not to soak up water. Cut them into chunks and cook in salted boiling water for about 30 minutes.
While they simmer, finely chop the onion, crush the garlic, and chop the parsley. Combine the olive oil and melted butter in a small bowl; this will be your cooking and brushing mix.
Once cooked, cool the mushrooms in cold water. When they’re easy to handle, cut them into thin slices or small cubes.
Warm a little oil in a pan and sauté the onion and garlic until soft and fragrant. Add the mushrooms, season with salt, black pepper, and thyme, and cook briefly, just until the flavors come together. Stir in the parsley, then set the filling aside to cool.
Preheat the oven to 350°F (180°C). Line a baking dish with parchment paper and brush it with the butter-oil mix. Lay down one phyllo sheet and brush it with butter mixture too.
Spread a layer of the mushroom filling on top, cover with another sheet, and brush again.
Roll or gently crinkle the layered sheets from the wide side, keeping the folds about two centimeters high so they fit nicely in the dish. Arrange the filled roll along one side of the pan, leaving room for the rest. Continue until all the filling and sheets are used. Brush the top with the remaining butter mixture.
Bake for 15 minutes, until the phyllo turns golden and crisp. Cover loosely with parchment paper if it browns too quickly.
Whisk the eggs and sour cream together, seasoning with salt and pepper. Pour this mixture evenly over the baked pie, sprinkle with sesame seeds, and bake again at 395°F (200°C) for another 20 minutes, until golden on top.
Let the pie rest for a few minutes after baking so the filling sets slightly. Slice and serve warm. Enjoy :)