Preheat the oven to 450°F. Line a baking pan with foil. Rub the olive oil all over the portobello mushroom, then season with salt and paprika. Roast for 20-25 minutes.
Meanwhile, make the olive salad. In a small bowl, combine the green olives, Kalamata olives and chopped giardiniera until the mixture is uniform. Set aside.
Assemble. Start with the top portion of the bagel. Spoon about ¼ cup of the olive salad mixture onto the bread. Top with roasted red pepper slice, followed by provolone cheese slices.
Finish by placing the cooled portobello mushroom on top, then topping with the "bottom" piece of the bagel. Flip and cut in half or into quarters if serving as an appetizer.