This took me about 20 minutes of play to achieve (although I've played the game plenty before, so know how it all works), so it should be fairly straight forward to achieve even if you're new to the game.
If you look at the wiki (or the XML file, or the in-game crafting menu) you will see that Diggle Nog can be crafted using a Porta-Still and combination of Aqua Vitae, a Diggle Egg and some Grog.
Therefore, to get the achievement you need to find all three craft items and a Porta-Still to combine them with. Craft them together and that's your achievement.
So, to minimize the time to get this achievement I'd highly recommend starting with the Alchemy skill so that you start with some Aqua Vitae (which will probably be in your quick bar, not your inventory) and a Porta-Still. You'll then need to find the other two items yourself, but they're fairly common items that you should be able to find somewhere on the first level.
Keep on the lookout for Drink Machines:
If you take Burglary as a skill, you can steal random liquor from drink machines, which may speed up your acquisition of the Grog. Otherwise, check each one you find to see if they're selling any.
Diggle Eggs are found frequently in Diggle Nests, which are small rooms containing a couple of Diggles and usually nests filled with eggs:
Best Answer
APA Deck (APA = Accelerated Production Achievement)
Options: Primal Bellow (spell mastery), Ruin Processor (7-mana creature)
My APA Deck maximizes your chances on this achievement with multiple combinations of Animist, Swell of Growth, and sacrificed Eldrazi Scions created by dead Blisterpods. It also reduces time - you don't have to mulligan with no lands in your hand if the other cards are right. Play Swell (or Primal) on an opponent's creature to gain spell mastery for Animist.
Possible sequences:
OR:
Contrary to what others have said, Nissa's Pilgrimage is not too slow. It can be played T3 even without spell mastery. Start with: 2x Blisterpod, 1x Swell, 1x Pilgrimage, and 1x Forest:
With the APA Deck, I was able to get the achievement 3 times in 4 hours. Of course, your mileage may vary. Good Luck!