To prepare the dough: In a stand mixer with a paddle attachment, cream the butter and sugar together at medium speed until fluffy. Add the peanut butter, eggs, and vanilla. Beat just until combined (1-2 minutes).
In a large bowl, sift and combine the flour, baking powder, baking soda, and salt. Slowly add to the peanut butter mixture, beating at low speed until well incorporated.
Refrigerate cookie dough for about 30 to 60 minutes to firm up.
Meanwhile, prepare the filling by blending its ingredients in a mixer until smooth. Cover and refrigerate.
Preheat oven to 375 degrees F (about 190° C). Line 2 to 3 large baking sheets with parchment paper or silpat, and set aside.
Using an ice cream scooper, scoop about 20 cookies and place onto the lined baking sheets (about 2-inch apart from each other) and press the top of each cookie with the palm of your hand to flatten.
Bake for 13 -15 minutes. Place baking sheet on a rack and let cookies cool enough to handle without breaking apart. Place 10 cookies face down and top each one with 2 to 3 tablespoons of the peanut butter cheesecake filling. Top each cookie with another cookie to form a sandwich. Set aside.
Make the chocolate drizzle: Microwave both the chocolate chips and heavy cream in a microwave-safe bowl at medium power (50%) for about 60 to 90 seconds and stir well. If necessary, heat again at 15-30 second intervals until chocolate is melted and smooth. Using a fork, drizzle melted chocolate over the sandwich cookies. Let drizzle set and enjoy!