Preheat oven to 350°F (175°C). Line baking sheets with parchment paper.
In a large bowl, cream together the butter and sugar until light and fluffy. Beat in the eggs one at a time, then stir in the vanilla extract.
In a separate bowl, whisk together the flour, baking soda, and salt. Gradually add the dry ingredients to the wet ingredients, mixing until just combined.
In a small bowl, combine the brown sugar and cinnamon.
Roll the dough into small balls. Flatten slightly, then sprinkle with the cinnamon-sugar mixture. Gently swirl the mixture into the dough.
Place cookies on the prepared baking sheets and bake for 8-10 minutes, or until the edges are lightly golden brown.
Let cool on baking sheets for a few minutes before transferring to a wire rack to cool completely.
In a small bowl, whisk together the powdered sugar, milk, and vanilla until smooth. Drizzle over cooled cookies.
Notes
Enjoy these delicious cinnamon roll sugar cookies with a glass of milk or a cup of coffee!