Using a hand mixer or a stand mixer fitted with the paddle attachment, cream the butter, cream cheese, and sugar until smooth. Add the vanilla extract and mix to combine. Gradually add the flour mix in, beating on low speed until cookie dough forms.
1 cup (226g) unsalted butter, 2 ounces cream cheese, 1 cup (200g) granulated sugar, 1 teaspoon vanilla extract
Transfer the cookie dough onto a clean piece of parchment paper or silicone mat. Divide into two parts and roll each out to a rectangle, about ¼" thick.
To make the filling, combine the melted butter, brown sugar, sugar, and cinnamon in a small bowl. Divide the filling evenly among the two rectangles and spread a thin layer across each one. Tightly roll each rectangle up into a log. The dough will be soft and sticky. Wrap each log in plastic wrap and refrigerate at least 2 hours.
When ready to bake, preheat oven to 350°F and line a baking sheet with parchment paper. Remove dough from the refrigerator and slice into ½ in. slices. Arrange on prepared baking sheet, about an inch apart, and bake for 14-16 minutes. Allow cookies to cool completely before removing from baking sheet and icing.
To make the icing, combine the powdered sugar and milk until you achieve a drizzling consistency. Spread or drizzle over cooled cookies.
1 cup (113g) powdered sugar, 3 tablespoons whole milk
Notes
To make these cookies gluten free, simply swap in a gluten free flour blend that contains xanthan gum (I use Bob's Red Mill 1 to 1).
A few cracks in the logs are ok. Don't stress too much about making them perfect!
If your icing is too thin, add more powdered sugar. If it's too thick, add more milk. Either way, add gradually (one tablespoon at a time).