Preheat oven to 350 degrees, with rack set in lower third. Butter a shallow 2-quart baking dish; set aside.
In a large bowl, whisk together eggs, vanilla, salt, and 1/2 cup sugar until combined; whisk in milk. Add bread, bananas, and raisins; toss gently to combine. Set aside to let bread absorb liquid, about 10 minutes, stirring occasionally.
Transfer mixture to prepared baking dish; sprinkle with remaining tablespoon sugar. Bake until a toothpick inserted in center of bread pudding comes out clean, 50 to 60 minutes. Let cool 10 minutes before serving.