Prepare the Dough: mix in a bowl the flour and salt. Add the oil and mix with a spoon. Slowly add the water (continue mixing with spoon) until it compacts into a homogeneous mass, don't be afraid to work it with your hands a bit and to add more flour if necessary. Form into a small disc, wrap it in cling film and keep in the fridge while you prepare the filling. Preheat the oven to 350 degre
Sauté de onions: in a large skillet, melt half a tablespoon of butter and the olive oil over high heat. Once the butter is bubbling add the sliced onions, after 5 minutes sprinkle with some salt and chopped rosemary and cover. Keep cooking until the onions are reduced and light brown in color, about 20 minutes, stirring every 5 minutes. Before removing the pan from the burner, turn the heat to high and add one tablespoon of vinegar. Sprinkle the onions with some pepper and transfer them into a bowl.
Sauté the mushrooms: in the same skillet where the onions were sauteed, melt half a tablespoon of butter and one tablespoon of neutral oil over high heat. Drop the mushrooms into the pan and sauté, cook until most of the liquid has evaporated (in my case there was about 2 tablespoons of liquid left in the pan). Add the tablespoon of vinegar and continue to cook for about 2 minutes. Sprinkle with salt, pepper and thyme leaves. Bring the onions back into the pan, reduce the heat to low and mix the cream into the onion-mushroom mixture, cook for about 4 minutes and remove from the heat.
Roll, assemble and bake: take the dough out from the fridge; over a clean surface dusted with flour roll the dough into a rough 1/4 inch circle and transfer onto a prepared baking sheet (you can either: butter the sheet, use parchment paper or a non stick mat). Leaving a 2 inch border, spread the vegetables onto the dough in an even layer and fold the edges over (I typically make 5 pleats in total, so the end product has a pentagonal shape). Place in the oven and bake for 25 - 30 minutes, taking it out half way through baking to brush the edges of the dough with the beaten egg yolk. Garnish with crumbled goat cheese and fresh thyme springs.