Pre-heat the oven to 400 degrees. Cover a large rimmed baking sheet, or appropriately sized ovenproof plate with foil. Toss mushrooms with oil, water and red wine vinegar; season with salt and pepper. Arrange the mushrooms stem side down; cover loosely with more foil. Roast until tender, 30 to 45 minutes.
Meanwhile brush the cut side of all the Kaiser roll halves with a little olive oil. Then set them oiled side down onto a grill pan, until lightly toasted.
When the mushrooms are ready remove the top layer of foil and turn each cap over so that the ribs are facing up. Carefully drain off any of the accumulated liquid. Fill each cap with some of the grated cheese. Set the caps under a broiler until the cheese is bubbly and beginning to brown.
Lay the toasted rolls out on a large surface or serving tray. Cover the tops of each roll with 5-6 whole basil leaves. Top the basil with a mushroom cap. The push 5 or 6 halved Kalamata olives into the melted-cheese. The cheese should act like glue keeping them from rolling off the "burger".
On the other side of the roll layer the red bell pepper and tomato. Invert the top onto the bottom of the burger.
This is a messy burger so I advise cutting it in half to make it easier to handle.