Make the Filling: In a medium bowl, mix butter, brown sugar, cinnamon, and salt with an electric mixer until fluffy (3-5 minutes).
Prepare the Topping: Combine white sugar and cinnamon in a small bowl; set aside.
Mix the Dough: Preheat oven to 350°F (175°C). Whisk together flour, baking powder, soda, salt, and cream of tartar in a medium bowl. In a large bowl, beat butter, brown sugar, and white sugar until fluffy. Add egg yolks and vanilla, mix well. Gradually add dry ingredients and combine.
Incorporate the Filling: Divide dough into thirds. Sprinkle one-third of the filling on each portion and fold it in, keeping swirls intact.
Form Cookies: Scoop dough into balls, roll in cinnamon sugar, and place on parchment-lined baking sheets.
Bake: Bake for 8-10 minutes until edges are golden and centers are soft.
Make the Icing: Beat cream cheese until smooth. Gradually add powdered sugar and milk until the icing is creamy.
Drizzle the Icing: Once cookies are cool, drizzle with cream cheese icing.