Preheat oven to 375 degrees F (190 degrees C). Grease baking sheets or line with parchment paper.
Stuff a split piece of pecan half into each date. Beat 6 tablespoons butter and brown sugar in a mixing bowl until creamy; beat egg yolks into mixture until thoroughly combined.
Whisk flour, baking soda, and baking powder together in a separate bowl and gradually mix into moist ingredients, alternating with sour cream. Stir dates into dough to coat and use 2 tablespoons to pick up 1 date plus enough dough to cover. Drop dough and date onto prepared baking sheet; use the 2 spoons to shape dough around each date to cover.
Bake in the preheated oven until cookie is golden brown, 7 to 9 minutes. Cool.
Melt 6 tablespoons butter in a saucepan and heat until butter is golden brown in color and gives off a nutty fragrance, about 2 minutes. Stir confectioners' sugar, vanilla extract, and milk into hot butter and mix until frosting is smooth. Working quickly, spread warm frosting onto each date cookie. Frosting will set quickly and become hard to spread as it cools.