Preheat the oven to 300. Clean and peel the puffballs, then chop them into 1 inch cubes. Put the chopped puffballs into the biggest stockpot with a lid you have, then add a cup or two of water so the puffballs don't scorch on the bottom. Cover the pot, then turn the heat on medium, stirring occasionally, until the puffballs are wilted and have started to give up their water. At this point, the puffballs will have turned varying shades of nasty looking blue and gray, don't worry, this is natural, it happens when puffballs are exposed to moist heat.
As the puffballs wilt and give up their water, more space will be created in the pan, if you have more mushrooms to add, do so and repeat the process, just make darn sure they don't burn on the bottom.
When the mushrooms are wilted and soft, begin transferring them to the bowl of a blender to puree. If you have difficulty getting the puree to be smooth, add some warm water until the blades of the blender begin to buzz it all up.
When you have a smooth puree, transfer it to a very wide pan that you've greased liberally with lard or oil. Put the pan in the oven and cook, whisking every 15 minutes or so, until most of the water has evaporated and the mixture is browned and caramelized. Eventually the puree will break and end up looking like cooked ground beef, don't worry this is normal.
When the puree is nicely browned and evenly caramelized, remove the pan from the oven and allow the puffball matter to cool to room temperature. When the butter has come to room temperature, put batches of it in a food processor and process it, adding small amounts of chilled butter, a tablespoon at a time until a velvety puree is formed.
Once you have a smooth puree, remove it to a container with a tight fitting lid, label, date, and refrigerate or freeze until needed. It's possible to over process the mixture and break it if the puffball mixture is too warm, or if it's processed for too long, if that happens, you can drain the butter from the mixture, cool the puffball puree, and repeat the process with some more chilled butter.