Roasted Tomatoes: In a bowl, combine oil, vinegar, garlic, thyme, salt and pepper. Mix well. Brush bottom of sheet pan with oil mixture. Evenly spread tomato slices on sheet pan in a single layer. Drizzle remaining oil mixture on top of tomatoes. Roast tomatoes at 400 degree F for 20 minutes until tomatoes start to caramelize.
Basil Puree: In a small pot, bring water to a boil. Place basil in a wire strainer dip in boiling water for 10 seconds. Remove basil from water and immediately shock under cold water for 10 seconds. In a food processor or blender, add basil. Pulse until finely chopped. Add garlic, olive oil and salt to food processor. Pulse until combined. Mixture will thicken.
Basil Mayonnaise: In a medium bowl, combine all ingredients, mix until well blended.
For Service: Evenly spread eggplant on sheet pan in a single layer, drizzle oil on top of eggplant, season with salt. Place eggplant on medium high grill. Grill for 1 to 2 minutes on each side. Place 2 slices cheese over top 3 slices of eggplant. Allow to melt. Split rolls. Spread 1 tablespoon basil mayonnaise on each, the heel and the crown roll. Place eggplant with melted cheese on top of heel of roll. Place 1/4 cup roasted tomatoes on top of eggplant. Close sandwich with crown of roll. Cut in half.