For the cookies: Stir together the flour and baking soda in a bowl. Cream the butter and both sugars on high speed in an electric mixer fitted with the paddle attachment until light, about 2 minutes. Add the peanut butter and beat until smooth. Add the eggs, one at a time, and beat until incorporated. Beat in the vanilla. Add the flour mixture and beat on low until just combined. Divide the dough in half, form into 2 rectangles, wrap in plastic wrap and refrigerate until firm, about 1 hour.
Cut a 12-inch peanut shape out of parchment to use as a template. Roll one piece of dough between two pieces of parchment to a rectangle a little bigger than the template. Remove the top piece of parchment and, using a pizza wheel or paring knife, cut the dough into a peanut shape. Reserve the dough scraps. Repeat with the remaining dough piece. Mold the scraps into a rectangle, wrap in plastic and chill along with the peanut shapes until firm, about 30 minutes.
For the filling: Meanwhile, preheat the oven to 350 degrees F. Beat the peanut butter and butter in a large bowl with an electric mixer on medium speed until combined. Slowly add the confectioners' sugar, about a 1/4 cup at a time, beating in between additions, until completely incorporated.
To build the cookie: Roll out the chilled rectangle of dough scraps into a large rectangle about 1/4-inch thick. Use a pizza wheel or sharp paring knife to cut long strips about 1/2-inch wide. Transfer the peanut-shaped pieces of dough on their pieces of parchment to 2 baking sheets. Lay the dough strips on top of one of the peanut-shaped pieces of dough in a crisscross, freeform pattern to mimic the lines on a peanut. Use a rolling pin to gently flatten and adhere the strips to the peanut.
Bake the cookies until lightly golden around the edges and dry on top, about 30 minutes. Cool completely on the baking sheets on a wire rack.
Turn over the cookie without lines so it sits flat-side up on a flat serving plate. Spread the filling evenly to the edges. Top with the second cookie, lined-side up. Use a serrated knife to cut into squares for serving.
When measuring flour, we spoon into a dry measuring cup and level off the excess. (Scooping directly from the bag compacts the flour, resulting in dry baked goods.)
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