Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
In a large bowl, cream together the softened butter, granulated sugar, and powdered sugar until fluffy.
Add eggs and vanilla, mixing well.
Mix in baking soda, cream of tartar, salt, and flour until a dough forms.
In a small bowl, mix melted butter, brown sugar, and cinnamon to make the swirl filling.
Take a small portion of dough, flatten it, add a bit of the swirl filling, fold and roll into a ball.
Place dough balls on prepared baking sheets and slightly flatten.
Bake for 9–11 minutes or until edges are set but centers are soft. Let cool.
Mix powdered sugar, milk, and vanilla to make icing. Drizzle over cooled cookies.