Spray 13x9-inch (3-quart) glass baking dish with cooking spray. In large bowl, beat eggs with wire whisk until foamy. Beat in milk until blended; beat in salt and pepper. Set aside.
Place bread cubes in baking dish. Sprinkle with mozzarella cheese. Pour egg mixture over top, pressing lightly to moisten bread. Using spoon, swirl pesto through mixture. Sprinkle Parmesan cheese over top. Cover with plastic wrap; refrigerate at least 8 hours but no longer than 24 hours.
Heat oven to 350°F. Remove plastic wrap; bake uncovered 40 to 45 minutes or until strata is puffed and knife inserted in center comes out clean. Let stand 5 minutes before serving. Cut into squares.