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 tsp in size. They should be small like chips. Freeze 10 minutes, until firm.
Cookies
Add 1 stick of butter to a skillet set over medium heat. Cook until the butter begins to brown, about 3-4 minutes. Remove from heat and transfer to a heatproof bowl. Freeze for 10-15 minutes, but no longer.
Preheat the oven to 350F. Line a baking sheet with parchment paper.
In a bowl, beat together the remaining stick of butter, the cooled browned butter, and the brown sugar until combined. Beat in the eggs, one at a time, until combined. Add the vanilla, beating until creamy. Add the flour, baking soda and salt. Fold in the frozen cinnamon chunks.
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.
Bake for 8 minutes, rotate pan and bake for another 1-3 minutes or until the cookies are just beginning to set on the edges but still doughy in the center.
Remove from oven and let the cookies cool on the baking sheet. They will continue to cook slightly as they sit.
Icing
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 a little more milk if needed to thin.
Spread or drizzle each cookie with icing.
Eat warm or let cool to set the icing. The icing will harden/dry enough to stack.
Notes
Can be stored in an airtight container for up to 4 days. I imagine that these cookies would freeze well although I have yet to try.The cookie dough balls can be frozen for up to 3 months. Just thaw and bake as directed.