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In the bowl of an electric mixer, cream the butter and brown sugar on medium-high speed until light and fluffy, about 4 minutes. At low speed, stir in the dry ingredients alternating with the milk mixture. Cover the dough with plastic wrap and refrigerate for about 1 hour.
Preheat the oven to 350 degrees F and line two baking sheets with parchment paper.
Shape the dough into a disk and sandwich between two pieces of floured parchment paper. Roll out to 1/4-inch thick. Cut out cookies using a 2-inch round cookie cutter. Gather up the scraps, roll again and cut out more cookies. Arrange the cookies about 2 inches apart on the prepared baking sheets.
Bake until the edges are lightly golden brown, 10 to 12 minutes. Let them cool completely on the baking sheets.
For the ganache: Put the chocolate in a bowl. In a saucepan, bring the cream to a boil. Remove from the heat and pour over the chocolate. Let melt for 1 minute without stirring. With a whisk, stir until smooth. Whisk in the butter. Cover and refrigerate until the ganache is cold but still spreadable, about 30 minutes.
For the marshmallow: Sprinkle the gelatin over 1/4 cup water in the bowl of an electric mixture and let bloom for 5 minutes.
Beat the gelatin mixture on medium high and slowly drizzle in the sugar mixture and vanilla. Continue to beat until the marshmallow topping cools and forms stiff peaks, about 15 minutes. Transfer to a pastry bag or resealable plastic bag if desired.
Preheat the broiler.
To assemble the cookies, spread about 1 teaspoon of the ganache on each cookie. Top with a piped swirl or dollop of marshmallow. Broil until the marshmallows turn golden, rotating as needed, about 1 minute.
Let cool and serve.