Place the bread cubes in a greased 2-qt. casserole; pour butter over top and toss to coat. In a medium bowl, lightly beat eggs; add milk, sugar, vanilla, cinnamon, nutmeg and salt. Stir in bananas. , Pour over bread cubes and stir to coat. Bake, uncovered, at 375° for about 40 minutes or until a knife inserted in the center comes out clean. , Meanwhile, for sauce, melt butter in a small saucepan. Combine sugar and cornstarch; add to butter. Stir in milk and corn syrup. Cook and stir over medium heat until the mixture comes to a full boil. Boil for 1 minute. Remove from the heat; stir in the vanilla. Serve warm sauce over warm pudding.