Preheat your oven to 375°.
In a large frying pan, add 1 tbsp of the oil and all of the onion. Cook over medium heat stirring often until the onions are caramelized. This should take 10-15 minutes. Add in the mushrooms and saute until they release their water. Add in the garlic and cook for an additional minute.
While the onions are cooking, brush a baking sheet with oil. Start layering the phyllo pastry. Brush each layer with oil before layering the next piece. Once you have layered all 10 pieces, fold over the edges as best as possible to create a crust. Brush with oil.
When the onions and mushrooms are done, layer evenly over the phyllo pastry. You don't need to leave much of an edge. Sprinkle the feta cheese evenly over the mushrooms and onions.
Bake in he preheated over for about 30 minutes or until golden brown.
While the tart bakes, toss the arugula in the balsamic vinegar.
Once the tart is out of the oven and has cooled for a few minutes, sprinkle the arugula on top. Slice the tart into squares and serve immeditely.