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Mix the lemon juice, olive oil, salt, pepper, and oregano together in a large bowl. Toss the mushrooms in the lemon juice mixture. Let the mushrooms marinate while the grill gets hot.
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Set your grill to medium heat. Once hot, add the mushrooms, top side down to the grill. Grill for five minutes on each side.
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Layer the mushrooms, sliced tomatoes, and sliced mozzarella. You can do smaller stacks or one big stack, whatever works best for the crowd you are serving. Drizzle the olive oil and balsamic vinegar over the stack/stacks along with salt, pepper, and fresh basil.