Place a 12inch skillet over mediumhigh heat. Add the olive oil and butter to the pan. Once the butter melts and starts to foam, add the diced onions and saute until translucent, about 3 to 4 minutes. Add the garlic and cook until fragrant, about 30 seconds. Add the mushrooms and cook, stirring often, until they have released most of their liquid, about 10 minutes. Add the thyme, oregano, chopped tomatoes, tomato paste and stock to the pan. Bring the contents of the pan to a boil and reduce to a gentle boil. Continue to cook, stirring occasionally until the liquid has reduced and thickened, about 20 minutes. Season with the salt and pepper, and add the basil and parsley to the pan.
In another large saute pan, over medium heat, heat the remaining 2 tablespoons of oil. Panfry the cheese until the crust is golden, about 2 minutes on each side. Remove and drain on paper towels.
To serve, spoon the mushroom ragu onto 4 serving plates. Arrange 2 pieces of the fried cheese on top of the mushroom ragu. Garnish with shaved truffles and a drizzle of truffle oil.