Roast the corn in their husks directly on oven rack at 500°F for 15 minutes. Cool slightly, then shuck the corn and remove silks.
Whisk together all remaining ingredients in large bowl. Submerge the corn in the yogurt mixture. Cover and refrigerate for at least 1 or up to 48 hours.
For the basil oil:
Heat the water to a boil in a saucepot over high heat. Add the basil leaves and submerge with a spoon for 10 seconds. Strain and immediately plunge into an ice bath. Transfer basil onto half of a clean towel with a slotted spoon; fold towel over basil and pat dry.
Transfer basil to a food processor. Add the oil and process for 20 seconds or until the basil is finely chopped but not puréed. Strain the basil oil through cheesecloth, if desired (discard or reserve the solids for another use). Makes about ¾ cup.
For the oven roasted tomatoes:
Toss the tomato quarters with oil, salt, and pepper in a bowl. Arrange tomatoes cut side down on a sheet tray and roast at 350°F for 45 minutes.
For the grilled eggplant:
Heat the garlic, oil, and oregano to a simmer in a saucepot over medium heat; simmer for 1 minute and remove from heat.
Brush both sides of eggplant slices with garlic oil and sprinkle with salt. Grill eggplant over high heat for 8 minutes or until tender, turning once. Cool; cut eggplant slices lengthwise in half.
Slice the corn off 1 ear of corn, keeping the kernels attached as much as possible. Heat the corn, 3-4 roasted tomato quarters and 4 pieces eggplant on a sheet tray at 350°F for 8 minutes or until heated through.
Fan eggplant on a plate and top with 1-ounce cheese and corn. Arrange the tomatoes around the eggplant and drizzle with basil oil.