Reconstitute the sun-dried tomatoes by soaking them in boiling water for 10 minutes. Remove the tomatoes from the water and chop them.
Slice the stems off of the mushrooms so they can lay completely flat. Slice each cap in half so you have 8 round mushroom slices. Carefully cut each slice into a heart shape with a cookie cutter.
Top half of the heart shaped mushrooms with 1 tablespoon of tomatoes and 1 tablespoon of cheese. Top with remaining mushroom slices. Brush the top side of each "sandwich" with olive oil.
Preheat a large, nonstick skillet or grill pan. Place the mushroom sandwiches in the pan carefully, oil side down, and cook for 2 minutes. Brush the top half with oil, flip, and cook for 2 minutes more. Season with salt and pepper, to taste, and serve.