To a wide, shallow bowl, add eggs and milk. Whisk.
Soak banana bread slices in egg mixture until completely soggy. Reserve on a plate.
In a pan, over medium to medium-high heat, melt butter. When butter is bubbling, add banana bread. Cover pan with lid. Cook until golden brown on both sides and cooked throughout (approximately four minutes on the first side and three minutes on the second side). If you have extra bananas, add a few slices to the pan and brown them as you cook the banana bread.