Mix together the egg, yolk, vanilla, rum, sugar, and half-and-half in a broad, shallow bowl. Put 2 tablespoons of the butter in a large nonstick skillet and turn the heat to medium-high. Put each slice of the bread in the batter and, when the butter foam subsides, cook the bread until nicely browned on both sides, in batches if necessary.

Meanwhile, put the remaining 2 tablespoons butter in another nonstick skillet over medium heat. When the foam subsides, cook the banana pieces, turning frequently and sprinkling them with the brown sugar, until lightly browned, 5 to 10 minutes.

Cover each piece of French toast with a portion of the bananas and sprinkle with confectioners’ sugar. Serve hot.