Preheat oven to 350°F.
Whisk together eggs, sugar, and salt in a large bowl. Add in milk and vanilla, then whisk to combine.
Divide the torn bread pieces between 12 greased regular-sized muffin cups. Top with custard mixture and let stand for 5-6 minutes, or until bread is soft and some of the custard is absorbed.
Bake bread puddings on center rack in oven until golden and a tester inserted in center comes out clean, 30 to 35 minutes. Transfer to a rack to cool slightly, about 5 minutes. Remove from pan, then cool on rack until warm, about 10 minutes more.
Meanwhile, prepare banana sauce: Cut bananas crosswise into 1/3-inch-thick slices. Melt butter in a 1-quart heavy saucepan over moderate heat, then add brown sugar and cook, while whisking, until smooth and well-combined, about 1 minute. Add bananas and a splash of juice or water, then simmer, stirring occasionally, about 2 minutes.
Serve bread puddings warm with a scoop of ice cream and banana sauce spooned over top.
Recipes adapted from: www.epicurious.com