Rinse eggplant; trim and discard stem end. Rinse and core tomato. Cut both into 1/2-inch slices. Brush both sides of each slice with olive oil.
Place eggplant and tomato slices in a single layer on two baking sheets. Broil 6 inches from heat (one sheet at a time), turning slices with a wide spatula, until eggplant is browned on both sides and tomatoes are soft, about 10 minutes total per sheet. Let stand until cool enough to handle.
In a food processor, whirl basil and garlic until finely chopped. Add eggplant and tomato and pulse until vegetables are coarsely chopped.
Spoon into a bowl and stir in lemon juice and salt and pepper to taste. Garnish top of dip with feta cheese and serve with pita breads for dipping (see note above), or cover dip and chill for up to 1 day.