In a medium bowl, mix eggs, eggnog, vanilla, cinnamon, nutmeg, salt, and rum, and whisk until well blended. Pour over bread slices.
In a small bowl, stir together the dark brown sugar, butter, and maple syrup until evenly combined. Smear over the bread slices. Cover with foil and refrigerate overnight.
In the morning, preheat the oven to 350°F.
Bake for 45 minutes, then increase heat to 400°F. Bake for another 15 minutes.
Remove from oven, and let stand 15 minutes before serving.
Recipe by Allons-Eat! at https://www.allonseat.com/gluten-free-eggnog-french-toast/