Pour 3/4 cup of the non-dairy milk into a measuring jar. Add the cornstarch, cocoa powder, sugar, and vanilla extract and stir until smooth.
Add the remaining 1 1/2 cup of the non-dairy milk into a pot and bring to a boil. Whisk in the cocoa cornstarch mixture and bring to a boil again while stirring constantly. Turn the heat down and cook it for one minute longer, continue stirring, while it becomes thicker.
Remove from the heat and stir in the chocolate until well combined. (Taste and adjust sweetness as needed).
Pour the pudding into the prepared cookie pie crust and cover tightly with foil (to avoid the pudding from forming a skin). Chill at least two hours to firm the pudding.