01 - In a saucepan over medium heat, melt the butter. Continue cooking, swirling often, until the butter foams and turns golden brown with a nutty aroma. Pour into a large mixing bowl and cool for 10 minutes.
02 - In a small skillet, melt 2 tbsp butter. Add diced apple, brown sugar, cinnamon, nutmeg, and lemon juice. Cook over medium heat, stirring, until apples are just softened and syrupy. Spread out on a plate to cool completely.
03 - To the cooled brown butter, add brown sugar and granulated sugar. Beat until combined. Add eggs and vanilla; mix until smooth. In a separate bowl, whisk together flour, baking soda, and salt. Gradually add dry ingredients to the wet, mixing until just combined.
04 - Divide dough in half. On parchment paper, roll each half into a 9x12-inch rectangle, approximately 1/4 inch thick.
05 - Mix brown sugar and cinnamon for the cinnamon sugar layer. Sprinkle half over each rectangle, leaving a 1/2-inch border. Evenly dot with cooled apple mixture.
06 - Starting from a long side, tightly roll up each rectangle into a log using the parchment paper as an aid. Wrap tightly and chill in the refrigerator for at least 1 hour, until firm.
07 - Preheat oven to 350°F (175°C). Line baking sheets with parchment paper. Slice each log into 1/2-inch thick rounds. Place cookies 2 inches apart on prepared sheets.
08 - Bake for 13–15 minutes, until edges are golden. Cool on the baking sheet for 5 minutes, then transfer to a wire rack.
09 - Whisk together powdered sugar, milk, and vanilla until smooth. Drizzle over cooled cookies.