2. Melt the butter in another saucepan. Stir in the brown sugar. Cook over low heat until the butter and sugar blend together. Add the milk and heat until the sugar dissolves. (The sugar will get hard when the milk is added; it will melt again as the milk heats.)
3. Whisk a few tablespoons of the hot milk mixture into the egg mixture. Gradually whisk in the rest of the milk mixture.
4. Over medium heat, whisk the combined mixture constantly until it comes to a boil. Immediately remove from the heat.
5. Pour into a bowl. (If you don't want a skin to form on the top of the pudding, sprinkle it lightly with granulated sugar.) Refrigerate. Cover with plastic wrap when cool.
6. Place 1 sliced banana into each of 8 serving bowls. Top with a few tablespoons of pudding.