Heat oven to 350°F. Line two cookie sheets with parchment paper. Set aside.In a small saucepan, cook/melt butter over medium heat, 8 to 10 minutes, stirring constantly until it becomes a deep golden brown. Remove from heat and let cool for about 10 minutes.
In a large bowl, break up cookie dough. Add 3 tablespoons of the cooled butter, the flour and egg; stir until well blended.Shape cookie dough into 32 (1 ¼-inch) balls. Roll balls in 3 tablespoons of the cinnamon-sugar mixture. Place 2 inches apart on the prepared cookie sheets.
Bake cookies for 10 to 12 minutes or until edges are golden brown. Carefully, remove from oven and let cookies cool on the cookie sheets.
Frosting: In a small bowl, whisk the remaining cooled brown butter, powdered sugar and half-and-half together until well blended and creamy.Frost cookies, then sprinkle each cookie evenly with remaining cinnamon-sugar. Enjoy!