Puree all the ingredients to form a sauce.
Rub mushrooms with oil. Season with salt and pepper. Place the mushrooms on the grill, rib side down, and cook for 5 minutes. Turn mushrooms and place green sauce over mushrooms. Add sliced goat cheese. Touch with a little more sauce. Close the top to the grill and smoke for 5 minutes more.
Lightly brush eggplant, zucchini and onions with olive oil and then season with salt and pepper. Grill for 3 minutes on each side. Grill tomato for 3 minutes on each side. While vegetables are hot, using a ceramic mold or bread pan lined with plastic wrap, make a layer of eggplant and then spread with 1-ounce of Jack cheese. Make a layer of red pepper and spread with 1-ounce Jack cheese. Make a layer of zucchini and spread with jack cheese. Make a layer of onion with jack cheese. Make a layer of green peppers and then a layer of goat cheese. Make another layer of eggplant and then another layer of goat cheese. Finally, make a layer of tomato and then a layer of basil. Lightly press, weight and allow to sit for 20 minutes. Turn upside down and release from the mold. Place left over goat cheese on top and garnish with basil.
Place tomatoes, meat side down, over medium hot grill for 8 minutes. Carefully remove and place in blender. Add remaining ingredients and blend until smooth.