Melt the butter in a small skillet over medium heat. Add apples, sugar and cinnamon to the skillet.

Cook, stirring occasionally, until the apples are tender, about 4-5 minutes.

Remove from heat and let cool completely. Drain any excess liquid from the cooled apples.

Preheat the oven to 375ºF (190ºC). Line two large baking sheets with parchment paper.

In a medium bowl, whisk together the flour, baking soda, salt, cinnamon and nutmeg. Set aside.

In a large bowl, beat the butter and brown sugar with an electric mixer at medium speed until light and fluffy.

Beat in the egg, egg yolk and vanilla extract, one at a time, until well combined

Stir in the flour mixture, just until combined.

Fold the the cooled apples into the cookie dough.

Divide the dough into 12 equal golf ball size portions and place them 3 inches apart on the prepared baking sheets.

Bake until golden brown, about 11-13 minutes. Let cool 1 minute on the baking sheet, then transfer to a cooling rack to cool completely.

In a small saucepan, cook the butter over medium heat until golden brown and fragrant. When brown specks start to form in the bottom of the pan, remove from heat.

Immediately transfer the browned butter to a small bowl. Whisk in the the powdered sugar, vanilla extract, salt and 1 tablespoon milk. Add more milk as needed to reach desired consistency.

Drizzle the icing over the cooled cookies.