Vegetable oil, cooking spray
3/4 cup (1 1/2 sticks) unsalted butter, cut into pieces
6 ounces semisweet chocolate, finely chopped
3 large eggs
1/2 cup packed light-brown sugar
1 teaspoon pure vanilla extract
1/2 teaspoon coarse salt
3/4 cup all-purpose flour
1/4 cup plus 2 tablespoons dark rum
Coarse sanding sugar, for rolling
1 Preheat oven to 350. Coat a 12-by-17-inch rimmed baking sheet with cooking spray; set aside. Melt butter and chocolate in a small heatproof bowl set over a pan of simmering water, stirring occasionally. Set aside.
2 Whisk together eggs, brown sugar, vanilla, and salt in a large bowl. Stir in chocolate mixture, then fold in flour. Pour batter into prepared baking sheet. Spread evenly with a rubber spatula. Bake until top is shiny and a cake tester inserted into center comes out with some crumbs attached, about 10 minutes. Let cool completely on a wire rack.
3 Break up brownie into small pieces; transfer to the bowl of an electric mixer fitted with the paddle attachment. With machine on low speed, pour in rum, and mix until crumbs start to come together to form a ball.
4 Shape into 1-inch balls, and roll in sanding sugar to coat. Transfer to a baking sheet; refrigerate, uncovered, until cold, about 2 hours. Serve chilled or at room temperature.