Chill a large metal mixing bowl and beaters in the freezer for 10 minutes. Line a 9-inch square metal baking pan with a parchment sling and lightly coat with nonstick spray.
Make the pudding first so it has time to set: In a bowl, whisk espresso powder into 2 tablespoons of the cold milk until dissolved. Add the remaining milk and the pudding mix; whisk briskly for 2 minutes. Cover and refrigerate until thickened, about 10 minutes.
Heat the oven to 325°F. Prepare the cookie dough according to the package directions using the softened butter and egg. Press evenly into the prepared pan, then refrigerate the pressed dough for 10 minutes to minimize puffing.
Bake the cookie base at 325°F for 20–22 minutes, until the edges are set and the center looks dry on top. Cool on a rack for 10 minutes, then gently press the surface with a flat spatula to level it. Let cool completely.
Whip the cream: In the chilled bowl, beat the heavy cream on medium-high to soft peaks. Gradually add powdered sugar and vanilla; continue beating to medium-stiff peaks. Refrigerate while you prepare the cheesecake cream.
For the cheesecake layer, beat cream cheese, granulated sugar, and fine sea salt on medium-high until smooth and fluffy, 2–3 minutes. Fold 1½ cups of the prepared whipped cream into the cream cheese mixture in two additions until no streaks remain. Stir in most of the mini chips, reserving a teaspoon for garnish if desired.
Assemble: Spread the cheesecake cream over the cooled cookie base. Chill 10 minutes to firm. Spoon the chocolate pudding over the cheesecake layer and smooth to the edges. Dollop the remaining whipped cream on top and swoop into an even layer. Sprinkle with the reserved mini chips.
Cover and refrigerate at least 3 hours (or overnight) before slicing. Use a hot, dry knife for the cleanest layers.