Place bread in a shallow baking dish large enough to hold slices in a single layer. Pour egg mixture over bread; soak 5 minutes. Meanwhile, gently heat rum and syrup in a small pot over low heat until just warm.

Heat 1 tablespoon butter in a large nonstick skillet over medium heat. Fry 2 bread slices until golden brown, 3 to 4 minutes per side. Transfer to a wire rack set over a baking sheet and place in the oven to keep warm. Repeat, using remaining bread and butter.