Preheat the oven to 375°F (190°C). Line two baking sheets with parchment paper. In a stand mixer fitted with the paddle attachment, cream together 1/2 cup of butter and brown sugar on medium speed until smooth. Add the egg and vanilla extract, and beat for 2 minutes, until well combined.
Add the all-purpose flour, baking powder, baking soda, and salt, and continue beating until well combined. Then add the sour cream and mix until fully incorporated. Gently fold in the chopped cashews with a spatula.
Drop tablespoons of dough onto the prepared baking sheet, making sure not to overcrowd the pan. Space the cookie dough portions about 2 inches apart to allow room for spreading while baking.
Bake for 8 to 10 minutes, or until lightly browned. Transfer the cashew cookies to a cooling rack to cool completely.
Meanwhile, prepare the frosting. Melt the butter in a saucepan over medium heat until lightly browned. Remove from heat and let the butter cool slightly. Stir in the half-and-half cream, vanilla extract, and confectioners' sugar. Beat the mixture until creamy and thickened. Frost the cookies and top each with a cashew half.