Preheat coals in a covered grill while preparing eggplant. Arrange preheated coals around a dip pan in a covered grill. Test for medium heat above pan.
Meanwhile, cut eggplant in half lengthwise. Using a grapefruit knife, hollow out eggplant, leaving a 1/4-inch-thick shell. Chop eggplant pulp (you should have about 2-1/2 cups).
In a large skillet cook chopped eggplant, mushrooms, sweet pepper, onion, garlic, and basil in hot oil over medium heat until nearly tender, stirring occasionally. Stir in tomato and salt. Spoon mixture into eggplant shells. Sprinkle with Parmesan cheese.
Tear two 24x18-inch pieces of heavy foil. Fold each piece in half to make two double thicknesses of foil that measure 18x12 inches. Place an eggplant half on each. For each packet, bring up two opposite edges of the foil piece and seal with a double fold. Fold remaining ends of each piece to completely enclose eggplant halves, leaving space for steam to build inside.
Place eggplant packets on grill over drip pan. Cover and grill for 20 to 25 minutes or until eggplant shells are tender. (Or, to bake, place the foil packets on a 15x10x1-inch baking pan. Bake the stuffed eggplants in a 375 degree F oven for 25 to 30 minutes or until tender.) Makes 4 servings.