Beat cream, brown sugar, and vanilla in the bowl of a stand mixer fitted with a whisk attachment on medium speed until starting to thicken, about 1 minute. Raise mixer speed to medium-high; beat until medium-to-stiff peaks form, 3 to 4 minutes.
Spread about 1 1/2 cups of the whipped-cream mixture on the bottom of a 9-in. pie plate. Top evenly with a single layer of cookies (all facing up), breaking cookies as needed to fill any gaps and create a solid layer. Repeat layers of whipped-cream mixture and cookies 2 additional times (for 3 layers total of each).
Spread top of pie with remaining whipped-cream mixture, using the back of a spoon or a small offset spatula to swirl. Loosely cover with plastic wrap, and refrigerate at least 6 hours or up to 12 hours.
Just before serving, remove plastic wrap and sprinkle chocolate chips around pie's perimeter. Pie may be wrapped lightly in plastic wrap and stored in refrigerator for up to 3 days.
All that whipped cream can get messy. For the nicest slices, use a large chef's knife—but be sure to have a bowl of hot water nearby for dipping the knife between cuts.