In a small saucepan, bring brown sugar, ½ cup water, molasses, ginger, cinnamon, and nutmeg to a boil over medium heat, stirring occasionally. Reduce heat to medium-low; cook until thickened, about 8 minutes. Let cool.

In a large saucepan, bring milk to a simmer over medium heat, whisking occasionally. Add chocolate and all but 2 tablespoons cooled syrup, whisking until smooth.

Place cookie crumbs in a shallow dish; place reserved 2 tablespoons syrup in another shallow dish. Dip the rim of each mug in syrup; coat in cookie crumbs. Serve hot chocolate in prepared mugs. Garnish with cinnamon sticks, if desired.