In a large mixing bowl, cream together the butter and sugars until light and fluffy, about 2 minutes. Beat in the egg and vanilla extract until combined. Gradually add flour, baking soda, and salt until the dough forms.

Divide the dough into two portions, shape into disks, wrap in plastic wrap, and refrigerate for at least 1 hour.

Mix together softened butter, granulated sugar, and ground cinnamon until well combined.

Roll out one disk of dough to ¼ inch thick. Spread the cinnamon filling, leaving a 1-inch border. Roll the dough tightly and pinch the seam.

Wrap the filled log tightly and refrigerate for an additional 30 minutes.

Preheat oven to 350°F (175°C). Slice the chilled log into ½-inch thick rounds, place on a baking sheet, and bake for 10-12 minutes.

Whisk powdered sugar and milk until smooth. Drizzle over cooled cookies.