Please, try on your own before using this cheat sheet.
The butter can be made in the Grease Abyss found deep below the ground in the caverns.
Part 1: First you have to go to the K.R.A.M.P.U.S. Churnery to the right side. Talk with the cat and insert the milk you acquired from the goats in Frost's courtyard in the open pipe.

Part 2: The pipe in the ceiling explodes and the machine breaks. Fixing the pipe and fixing the machine are two separate things.

Fixing the pipe: From the entrance head left until you get to a crossroad. Head down to the Power Station and find a P. Nocchio - he's got a "Discharged Repair Rod". Continue down this path until you get to a room that shoots lightning. You'll have to loose almost all your life to the lightning before the rod gets charged. You can now fix the pipe.

Fixing the machinery: From the entrance to the Abyss head left till you get to the crossroad. Now you go towards the Laboratory. If Part 1 and 2 have been completed the entrance should be open. *Be warned that there is currently a bug in this room that makes the scenario freeze after the first door has been opened*. This can be solved by loading to another save and trying again, sorry. We'd also like to give a friendly reminder that you can't escape the room at any time before the machine has been fixed. Completing the scenario will fix the problem.

Now all of this has been done you can now make butter again. Head down to the K.R.A.M.P.U.S. Churnery again and insert milk in the insertion pipe next to Krampus. Reap your milk and head on to the next challenge.