In a small bowl, whisk the oil, vinegar, salt, and pepper together; set aside.
Moisten a paper towel and wipe the mushrooms to remove dirt.
Cut out the stems and the gills; place the mushrooms in a glass dish, round side up; pour the oil mixture over the mushrooms, rubbing it in with your fingers; turn the mushrooms and lightly sprinkle the insides with salt and pepper; marinate for 10 minutes.
Remove the mushrooms from the marinade and place on a baking sheet, rounded side up.
Bake 10-12 minutes or until they begin to brown and shrink a bit.
Turn the mushrooms and bake 10-12 minutes, or until some of the liquid has surfaced and the mushrooms have browned further.
Remove from oven; let sit until cool enough to handle.
Mix the cheeses and oregano together in a bowl.
Scatter the cheese mixture evenly over the inside of 4 mushroom caps, followed by a tomato slice; sprinkle with salt and pepper to taste, and place the remaining 4 mushroom caps on top, rounded side up.
Heat a large nonstick skillet over medium heat for 2 minutes; put the mushroom sandwiches in the skillet (in batches if needed), cover, and cook 3-5 minutes, or until the undersides are dark brown and the cheese has begun to melt.
Uncover, and turn the mushroom sandwiches, pressing firmly with a spatula to flatten them slightly.
Cover and cook 2-3 minutes or until the undersides have turned a shade or two darker and the cheese has melted completely; let cool slightly and cut in half; serve.