Whisk together first 5 ingredients. Place portobellos in a shallow dish, and rub with marinade. Pour any extra over the top and let sit for 30 minutes.
Preheat grill to medium-high. Brush onion and tomato with oil. Grill onion and mushrooms for 3-5 minutes per side, until tender and slightly bronzed. Grill tomato for 3 minutes on one side.
Turn mushrooms gill-side up and lay the mozzarella on top. Grill for another 3-4 minutes, until cheese is melted.
Toast the buns on the grill for 1 minute and spread both top and bottom with the Faux Aioli. Top each bun with two mushrooms, 1 onion slice and a tomato.Serves 4 Prep time: 45 minutes Cook time: 15 minutes