Place the cookies in a food processor and pulse until finely ground into crumbs.
Add brown sugar, cinnamon, ginger, cloves, and salt. Pulse to combine.
With the processor running, drizzle in the oil until the mixture starts to clump and become pasty.
Add milk, 1 tablespoon at a time, and process until smooth and creamy. Scrape down sides as needed. Adjust milk for desired consistency.
Transfer to an airtight jar. Store at room temperature for up to 2 weeks, or refrigerate for longer shelf life (let come to room temperature before using if refrigerated).
Tips
For a vegan version, use plant-based milk and ensure cookies are vegan.
If the butter is too thick, add a little more milk; if too thin, add more cookie crumbs.
Try adding a pinch of nutmeg or cardamom for extra warmth.