Preheat the oven to 350 degrees F. Coat a rimmed baking sheet or a jelly roll pan with cooking spray.
Combine the butter and chocolate in a small metal or glass bowl set over a pan of simmering water. Melt the chocolate, stirring occasionally.
Whisk the eggs, dark brown sugar, vanilla, salt and ginger in a large bowl. Stir in the melted chocolate and then the flour. Pour the batter into the prepared pan and spread with a spatula. Bake until a toothpick inserted into the middle comes out mostly clean with a few crumbs, about 10 minutes.
Use a spoon to scoop the brownie into a bowl and beat together with the rum until crumbs start to come together to form a ball. Scoop with a cookie scoop to make 1-inch balls and roll between your hands. Roll in sanding sugar and refrigerate until firm, about 2 hours.