Spray a 9x13-inch baking pan with cooking spray. Arrange bread slices in baking pan in a single layer. Whisk half-and-half, eggs, vanilla extract, cinnamon, and salt together in a bowl; pour over bread slices. Cover with plastic wrap and refrigerate, 8 hours to overnight. Preheat oven to 350 degrees F (175 degrees C). Remove plastic wrap and top bread slices with bananas. Combine brown sugar, butter, and corn syrup in a bowl; drizzle over bread slices. Bake in the preheated oven until golden brown and set in the center, about 40 minutes.