According to the recipe, Stewart uses homemade chocolate chip cookies and a combination of sweetened heavy cream and mascarpone to serve as the foundation.
No matter what cookies you use, they can't be too soft or too thick, or the heavy cream will have trouble doing its job. Nilla Wafers, Oreos, and graham crackers are good choices.
After layering cookies and whipped cream, cover with plastic wrap and refrigerate overnight. The results will be a moist texture with a firm structure.