In a standing mixer fitted with the paddle attachment, or with an electric mixer, beat butter for 30 seconds. Add granulated sugar and salt and beat for an additional 2 minutes on medium, scraping bowl as needed.
Beat in 1 egg and vanilla. Beat in flour. Wrap dough in plastic wrap and chill until dough is easy to handle, about 30 to 60 minutes.
Remove dough from plastic wrap and place on floured surface. Roll out dough to a 15 inch by 10 inch rectangle. Lightly beat 1 egg and brush surface of dough. In a small bowl, stir together brown sugar and cinnamon. sprinkle mixture over dough.
Roll dough up from the long side. Place dough roll on a baking sheet, cover and freeze until firm enough to slice, about 30 minutes.
Preheat oven to 375 degrees. Prepare cookie sheet with a piece of parchment paper.
Using a serrated knife, cut dough roll into 1/4 inch slices. Place slices on prepared baking sheet, about 2 inches apart. Bake until edges are light brown, about 8 to 10 minutes. Remove from sheet and cool on wire rack.