Stir together brown sugar, cinnamon, melted butter, and flour for the filling. Portion into small discs and freeze for at least 30 minutes.
Cream butter, brown sugar, and granulated sugar until fluffy. Beat in eggs one at a time, then vanilla.
Whisk flour, baking soda, and salt together, then mix into the wet ingredients until just combined.
Scoop dough, flatten, place a frozen filling disc in the center, top with more dough, and seal into a ball.
Chill assembled dough balls for at least 30 minutes.
Preheat oven to 350°F. Bake for 11 to 13 minutes, until edges are set and centers look slightly underdone.
Cool on the baking sheet for 5 minutes, then transfer to a wire rack to cool completely.
Whisk together frosting ingredients and spread over cooled cookies.