Combine first 4 ingredients in the bottom of a large food storage bag. Add mushroom caps, seal bag and shake bag to coat caps, then let the mushrooms marinate for 15 minutes.
Grill mushrooms on indoor electric grill, starting with caps down, turning to caps up, 6 minutes on each side. Pan-fry in large nonstick skillet for same amount of time. Season caps while they cook on each side with grill seasoning or a little salt and pepper. Melt the smoked cheese over mushroom caps in the last minute of grill time.
To make the red pepper paste, combine the roasted red peppers, oil, parsley, garlic, salt, and pepper in food processor and pulse grind into thick sauce.
To serve, place the grilled mushroom caps topped with melted cheese on the bun bottoms. Top with spinach and onion. Spread bun tops with roasted pepper paste. Serve with Tomato and Onion Salad and Pasta Salad with Ricotta Salata and Broccolini.