Preheat oven to 350°F (175°C) and line a baking sheet with parchment paper.
Cream together softened butter, 1/2 cup brown sugar, and granulated sugar in a large bowl until fluffy.
Incorporate the egg and vanilla extract, mixing until smooth.
Whisk flour, baking soda, 1 teaspoon cinnamon, and salt in a separate bowl, then gradually stir into the wet ingredients.
In a small bowl, combine the filling ingredients (1/4 cup brown sugar, 1/2 teaspoon cinnamon, and melted butter) to form a thick paste.
Portion dough into 2-tablespoon balls, create an indentation in the center, insert 1 teaspoon of filling, and pinch the dough to seal.
Arrange on the baking sheet 2 inches apart and bake for 10 to 12 minutes until edges are lightly browned.
Cool cookies on the sheet for 5 minutes before transferring to a wire rack.
Whisk powdered sugar, milk, and maple extract until smooth, then drizzle over completely cooled cookies.