Chewy Frosted Cinnamon Swirl Snickerdoodles

1. To make the cinnamon swirl. In a bowl, mix the butter, brown sugar, and cinnamon. Line a small baking sheet with parchment. Crumble/tear the cinnamon mix over the baking sheet, creating small chunks, no more than 2 teaspoons in size. They should be small like chips. Freeze 10 minutes, until firm.
2. To make the cookies. Add 1 stick butter to a skillet set over medium heat. Cook until the butter begins to brown, about 3-4 minutes. Remove from the heat and transfer to a heatproof bowl. Freeze 10-15 minutes, but no longer.
3. Preheat the oven to 350° F. Line a baking sheet with parchment paper.
4. In a bowl, beat together the remaining stick of butter, the cooled browned butter, and the brown sugar until combined. Beat in the eggs, 1 at a time, until combined. Add the vanilla, beating until creamy. Add the flour, baking soda, and salt. Fold in the frozen cinnamon chunks. 
5. Mix the sugar and cinnamon in a bowl. Roll the dough into rounded tablespoon size balls and then roll through the cinnamon sugar. Place 3 inches apart on the prepared baking sheet. 
6. Bake for 8 minutes, rotate the pan and bake another 1-3 minutes or until the cookies are just beginning to set on the edges but still doughy in the center. Let the cookies cool on the baking sheet. They will continue to cook slightly as they sit. 
7. Meanwhile, make the icing. Melt the cream cheese in the microwave, stirring until smooth, 15-20 seconds. Whisk in the milk and vanilla. Add the powdered sugar, whisking to combine. Add milk, if needed, to thin. Spread or drizzle each cookie with icing. Eat warm or let cool and store in an airtight container for up to 4 days. The icing will harden/dry enough to stack.