In a heavy-bottom saucepan, bring heavy cream (1 3/4 cups) to a simmer, watching carefully to only bring the cream to a simmer, not boil. You will see small bubbles forming around the edges, this is when you know you've reached the correct temp.
Continue to simmer for 30 minutes, and whisk occasionally to incorporate the bubbles into the cream.
After 30 minutes, remove the saucepan from heat, and whisk in the erythritol (1 1/4 cups), butter (2 tablespoons), and vanilla extract (1 teaspoon).
Sprinkle xanthan gum (1/2 teaspoon) over the top of the cream. Whisk aggressively until xanthan gum is fully dissolved. (Xanthan gum has a tendency to clump when it comes into contact with the liquid, so spreading it out over the top will help to prevent this.)
Pour reduced cream into a large bowl, and place in the fridge for 1 hour, or until it comes to room temp.
While the reduced cream is cooling, make the cookie crumbles (skip to the section below).
Once the reduced cream and the cookie crumbles have cooled to room temp, remove them from the fridge. Whip the reduced heavy cream with an electric mixer for 3 minutes (until it's light and frothy).
Use a rubber spatula to stir the cookie crumbs into the reduced cream until well incorporated.
In a second large bowl, whip the remaining heavy cream (2 cups) with an electric mixer until you've formed soft peaks.
Gently combine the whipped cream with the reduced cream, keeping the combo light and aerated, but ensuring the ingredients are well combined.
Transfer the mixture to an insulated tub or container and freeze for 4-6 hours.