1. Preheat the oven to 350°F. Grease a 9-inch by 13-inch pan and set aside.
2. In a saucepan over medium heat, melt the butter and brown sugar, stirring constantly until smooth. Remove from the heat, and stir in the peanut butter, until creamy and combined. Set mixture aside to cool.
3. In a large bowl, whisk together flour, baking powder, baking soda, and salt.
4. Add the eggs, one at a time, to the peanut butter mixture, mixing well after each addition. Add the vanilla extract and stir to combine. Pour the mixture into the prepared dry ingredients, and stir until batter is smooth. Fold in the diced apples.
5. Transfer the batter to the prepared pan, spreading the batter evenly to the edges of the pan. Bake for 30 minutes or until the top is golden and the edges are slightly browned. Remove bars from oven and place them on a wire rack to cool.
6. While the bars are cooling, make the peanut butter glaze. In a medium bowl, combine the sifted powdered sugar, milk, peanut butter, and vanilla extract. Whisk until smooth. Drizzle the glaze over the cooled bars. Cut into squares and serve.