1. To make the cinnamon swirl, in a bowl, mix 2 tablespoons of butter, 1/2 cup of dark brown sugar, and 2 teaspoons of cinnamon. Line a small baking sheet with parchment paper. Crumble or tear the cinnamon mix over the baking sheet into small chunks, no larger than 2 teaspoons. Freeze for 10 minutes until firm.
2. For the cookies, add 1 stick of butter to a skillet over medium heat. Cook until the butter starts to brown, about 3 to 4 minutes. Remove from heat and transfer to a heatproof bowl. Freeze for 10 to 15 minutes, ensuring it doesn’t freeze solid.
3. Preheat your oven to 350 degrees Fahrenheit and line a baking sheet with parchment paper.
4. In a large bowl, beat together the remaining 1 stick of butter, the cooled browned butter, and 3/4 cup of dark brown sugar until well combined. Beat in the 2 large eggs one at a time, mixing until fully incorporated. Add 2 teaspoons of vanilla, beating until creamy.
5. Add the 2 1/4 cups of all-purpose flour, 1 teaspoon of baking soda, and 1/2 teaspoon of kosher salt to the mixture. Gently fold in the frozen cinnamon chunks.
6. In a separate bowl, mix 1/3 cup of granulated sugar with 2 teaspoons of cinnamon. Roll the dough into rounded tablespoon-sized balls, then roll through the cinnamon sugar mixture. Place the balls about 3 inches apart on the prepared baking sheet.
7. Bake for 8 minutes, then rotate the pan and bake for an additional 1 to 3 minutes until the cookies are just beginning to set on the edges but still doughy in the center. Allow the cookies to cool on the baking sheet; they will continue to cook slightly as they cool.
8. While the cookies are cooling, make the icing by melting 2 ounces of cream cheese in the microwave, stirring until smooth, about 15 to 20 seconds. Whisk in 2 to 3 tablespoons of milk and 1 teaspoon of vanilla. Then, add 1 1/2 cups of powdered sugar, whisking until well combined. Adjust with more milk as needed for consistency.
9. Spread or drizzle the icing over each cookie. Enjoy them warm, or let them cool completely and store in an airtight container for up to 4 days. The icing will harden enough to stack the cookies.