1. Sprinkle eggplant with salt and let sit for 30 minutes. Pat dry.
2. Brush the caps of each mushroom with a clean sponge. Remove and discard the stems. Slice mushrooms.
3. Heat ¼ cup of olive oil and butter in a saucepan. Add mushrooms, 1 tablespoon of garlic and wine and cook over medium heat for 8 minutes, until they are tender and begin to release their juices, stirring often.
4. Add in eggplant and mix making sure the eggplant gets coated with the juices. Cook for 2 more minutes or until eggplant have begun to soften. Set aside.
5. Preheat grill to medium heat.
6. Combine remaining olive oil and garlic in a small bowl and let sit for 10 minutes (or more, you want the oil to get the garlic flavor).
7. Roll out pizza dough and place on the grill, leaving grill cover open. Cook for about 2 minutes or until bottom of crust has begun to brown a bit (cook time depends on how hot your grill gets, mine gets pretty hot).
8. Flip pizza dough on the grill. Brush with the garlic olive oil and sprinkle ¼ cup of mozzarella (you can take the dough off the grill if its too hot to do this).
9. Close grill and cook for 1-2 minutes or until mozzarella has begun to melt. Spread sautéed eggplant and mushrooms on the dough and top with sliced tomatoes and remaining mozzarella.
10. Close grill and cook for 4 – 6 minutes or until mozzarella is melted and bottom of crust is browned.