Prepare charcoal or gas grill or traditional smoker for Indirect smoking on Low Heat. Use one layer charcoal for smoker and place food on top grate. For best results for this recipe, use hickory or mesquite wood chips or chunks.
Place tomatoes on cooking grate cut side up of fully smoking grill or smoker. Smoke 30 minutes to 90 minutes over charcoal or 10 to 12 minutes over gas; remove. Chill tomatoes in refrigerator and drain.
Meanwhile, spray bread with olive oil. Place bread oil side down on cooking grate. Grill Directly 2 to 3 minutes or until toasted; remove.
Spread toasted side of each bread slice with mayonnaise; sprinkle with equal amounts of basil. Remove skn which ill just fall away. Cut each 1/2 tomato into 2 slices; place 2 tomato slices over basil on each bread slice. Sprinkle tomatoes with salt and pepper. Top each sandwich with 2 slices of goat cheese.