Peel and chop the eggplant into rough 2-inch pieces. Toss the eggplant with 1 teaspoon of salt and let sit in a colander for 15-20 minutes to drain. Dry the eggplant pieces with paper towels.
Heat 3 tablespoons of olive oil in a large skillet. Add half the eggplant pieces. Cook over medium high heat turning the pieces so they cook evenly. Once cooked, transfer them to a bowl. Repeat with the remaining eggplant pieces.
Let the eggplant cool to room temperature.
Preheat the oven to 275℉. Butter six ramekins (5-to-6-ounce capacity). Place them on a baking sheet.
Put the cooled eggplant, ricotta, parmesan, eggs and basil in the bowl of a food processor fitted with the metal blade. Pulse 8 - 12 times until the mixture is smooth with some texture. Adjust the seasoning to taste. Divide the mixture evenly between the buttered ramekins.
Bake in the oven for 30 minutes until the mixture is lightly set. Let the timbales rest for 10 minutes.
To serve, run a knife around the edge of the eggplant mixture then invert the mold onto a serving plate. Garnish with some of the tomato sauce, a slice of mozzarella, cherry tomatoes and basil. Drizzle with olive oil and season with black pepper.