Combine marinade ingredients. Place caps into a baking dish and pour marinade over top. Use a brush to make sure all caps are covered with the marinade. Set aside and let marinate up to 2 hours.
Remove the portobellos from marinade (do not discard marinade) and place gill side down on a preheated gas, electric or charcoal grill over medium-high heat. Grill 3 to 4 minutes per side. Top each cap with cheese 2 minutes to melt before they are finished grilling.
In a medium bowl combine 1/2 cup mayonnaise and 2 tablespoons of the marinade.
Spread the flavored mayonnaise onto each bun and top with portobellos. Serve with sliced onion, lettuce and tomato.