Heat grill to medium-low. Combine Parmesan, thyme and salt and pepper in a small bowl.
Brush mushrooms with oil on both sides and season with salt and pepper to taste. Place the mushrooms, cap side down and cook until golden, about 4 to 5 minutes. Turn over and continue grilling until soft, about 5 to 6 minutes longer. When the mushrooms are cooked through, carefully divide the cheese mixture over the top of the mushrooms. Close the cover of the grill and cook until the cheese has melted. Remove from the grill and drizzle each mushroom with a teaspoon of white truffle oil.
In a saute pan, drizzle olive oil and saute the oyster mushrooms until golden brown and tender. Season with salt and pepper.
Place on top of portobellos and serve.