Preheat oven to 375ºF.
Rub the inside of each cup of a 6-cup muffin tin with butter and set aside.
In a medium-size mixing bowl, whisk together the eggs, milk, cinnamon, vanilla and nutmeg.
Cut a slit in the corner of each slice of bread from the middle of the slice to the edge. Dip each slice of bread in the egg mixture then lightly press it into a cup of the muffin tin, overlapping the pieces where you cut the slit to make it fit into the cup without tearing.
Bake the French toast cups for 12-14 minutes, until they're light golden brown.
While the cups are working, place a small saucepan over medium-low heat with the maple syrup and cinnamon stick. Keep the syrup warm and stir it occasionally while finishing up everything else.