Heat oven to 425° F. Generously spray 13x9-inch baking pan with no-stick cooking spray. Whisk together egg, sour cream and milk in large bowl. Whisk in cornbread mix and pepper until smooth. Stir in olive oil just until blended. Pour into prepared pan. Bake 12 to 15 minutes or until golden brown. Cornbread will be thin.
Cut tomatoes into medium-thick slices. Drain on paper towels. Stir together cheeses, mayonnaise, basil and chives until combined.
Spread about half the cheese mixture over cornbread. Arrange tomato slices over cheese in overlapping rows. Salt and pepper to taste. Place spoonfuls of remaining cheese mixture on tomato slices. Return to oven for 5 minutes or until cheese begins to melt. Turn on broiler. Broil until cheese is bubbly and lightly browned, 1 to 2 minutes. Cut into squares. Serve warm or at room temperature.