Preheat oven to 350 degrees F. Line two baking sheets with parchment paper.

Cream together ½ cup butter with ¼ cup peanut butter and ¼ cup granulated sugar until light and fluffy, about 3 minutes. Add 1 teaspoon vanilla extract, 1 ¼ cup all-purpose flour, ¼ teaspoon salt, and 1 teaspoon dry milk powder and stir just until combined. Cover and chill the dough for 30 minutes or until firm.

Dust a clean work counter with flour. Roll out the cookie dough until ⅛ inch thick, making sure to continuously dust the counter and rolling pin with flour. Cut out 2 inch-circles and carefully transfer the cookies onto the baking sheet, leaving ½ inch gap in between. Bake the cookies for 7-8 minutes or until pale golden. Cool the cookies completely on a wire rack.

Make the filling. Beat together 5 tablespoon creamy peanut butter with 1 ¼ cups powdered sugar and 3 tablespoon water until well combined. Add an additional tablespoon of water if the frosting is too thick. Transfer the peanut frosting into a piping bag. Flip over half of the cookies and pipe on the frosting. Stack another cookie on top to make sandwich peanut butter sandwich cookies.

Make the strawberry icing. Beat together ¼ cup powdered sugar with 1 tablespoon strawberry jam and 1 teaspoon water. Drizzle the icing on the cookies and serve.