In a mixing bowl, whisk together the eggs, sweetened condensed milk, and vanilla extract until fully combined. Set the mixture aside. Slice the banana bread into 4 thick, even pieces.
Heat the butter in a large skillet over medium heat. While the butter is melting, dip each slice of banana bread into the egg mixture, ensuring both sides are evenly coated. Place the slices into the skillet and cook, turning once, until both sides are golden brown.
Just before serving, lightly dust the French toast with confectioners' sugar for a finishing touch.