Line a 9-inch square pan with a parchment sling. Chill a metal mixing bowl and beaters in the fridge for 10 minutes. Preheat the oven to 325°F.
Stir together the peanut butter cookie mix with the add-ins listed on your box (commonly an egg and oil) until a soft dough forms. Press evenly into the lined pan, docking the surface a few times with a fork.
Bake the crust at 325°F for 18–22 minutes, until the edges look set and lightly golden. Cool on a rack for 10 minutes, then gently press the surface with a flat spatula to level. Slide the pan into the freezer for 10 minutes to speed cooling.
Meanwhile, make the whipped cream: In the chilled bowl, beat the heavy cream with 2 tablespoons powdered sugar and 1/2 teaspoon vanilla to medium-stiff peaks. Refrigerate to keep it cold.
For the peanut butter cheesecake, beat the softened cream cheese until completely smooth. Mix in the peanut butter, 1/2 cup powdered sugar, 1 teaspoon vanilla, and salt until fluffy, about 1–2 minutes. Fold in about two-thirds of the whipped cream (reserve the rest for topping) until no streaks remain.
Spread the peanut butter cheesecake over the cooled crust. Chill 10–15 minutes to help it firm up before adding the next layer.
Whisk the pudding mix with cold milk and espresso powder for 2 minutes until thick and smooth; let stand 5 minutes to finish setting. Spoon over the cheesecake layer and smooth. Tap the pan lightly on the counter to release air bubbles. Chill 15 minutes.
Dollop the reserved whipped cream on top and swirl to cover. Refrigerate at least 2 hours (or overnight) for the cleanest slices.
Warm 3 tablespoons peanut butter in the microwave for about 20–30 seconds until pourable. Drizzle over the whipped cream and finish with chopped peanuts or chocolate shavings, if using. Lift out by the parchment and cut into 12 bars.