Cut the crusts off the bread slices and cut each slice into quarters to make squares, about 1½ inches on each side.
Whisk the egg and yolk in a medium bowl. Whisk in the salt, sugar, and vanilla. Gradually whisk in the half-and-half and the milk. Pour the mixture into a shallow baking dish. Working in batches if necessary, place the bread pieces in the egg mixture and let them soak, then turn them and soak on the other side.
Melt the butter in a skillet until it is foamy and very hot. Working in batches, brown the soaked bread on both sides.
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