To make the syrup: In a medium saucepan, combine brown sugar and cinnamon sticks with 1 cup water and bring to a boil. Reduce heat to low and simmer uncovered until the sugar has completely dissolved, stirring occasionally, about 10 minutes.
Remove the pan from the heat and let the syrup cool. Discard the cinnamon sticks. (Leftover syrup may be kept refrigerated for a week.)
Combine the syrup, bourbon, apple cider, lemon juice and orange juice in a pitcher. Refrigerate for at least an hour.
Strain the chilled punch into a punch bowl. Add 4 cups of ice and club soda. Garnish with orange wheels. Generously grate nutmeg on top. Serve over ice.