Preheat the oven to 275 degrees F (135 degrees C). Grease a 10x15 inch jellyroll pan.
In a large bowl, cream together 1cup of butter, white sugar and brown sugar until light and fluffy. Mix in the egg yolk and 1 tablespoon of vanilla. Combine the flour, cardamom and salt; stir into the batter until it forms a soft dough. Spread evenly in the prepared pan. Brush the top with egg white and sprinkle pecans over the top.
Bake for 1 hour in the preheated oven. Prepare the frosting while the cookies bake.
Melt the remaining butter in a saucepan over low heat until it starts to brown. Remove from the heat and stir in confectioners' sugar until smooth. Stir in enough milk to achieve a drizzling consistency. Drizzle frosting onto the bars when warm. Cool, then cut into bars.
