There's a total of 12 beasts you can breed. See below for details regarding the questline and where to capture the beasts:
During the main story quest "The Elf, The Nab-Sack, and the Loom" you'll be introduced to the "Nab-Sack" for capturing beasts and a Vivarium to house them. Shortly after that Deek will give you a further two side quests “The Plight of the House-Elf” & “Foal of the Dead.” During the "Foal of the Dead" quest you'll obtain the breeding pen and conjure it into a Vivarium. Now that we have a pen placed, ensure you have a Male & Female beast of the same species in that Vivarium. Interact with the book in front of the breeding pen entrance to start the breeding process. The pair will take 30 minutes to produce an offspring.
Notes:
- If you visit a Beast Den that only has one sex of beast available to capture. Just return at a later time and hopefully it's been swapped out for the one you need (e.g. Graphorn & Unicorn).
- Utilize the spell "Arresto Momentum" to stop the beast you wish to capture from running away. It makes it a lot easier to catch them with the Nab-Sack.
- You can place up to 4x Breeding Pens in each Vivarium. So in theory, if you had enough Moonstone you could breed all 12 species at the same time.
- The achievement tracking progresses when you place a beast inside the Breeding Pen. Achievement pops when you place in the 12th.
Beasts:Puffskein - Den discovered & captured during the "The Elf, The Nab-Sack, and the Loom."Jobberknoll - Den discovered & captured during the "The Elf, The Nab-Sack, and the Loom."Mooncalf - Den discovered & captured during the "The Elf, The Nab-Sack, and the Loom."Common:Diricawl - Found at any of the 5 Dens. South West of Marunweem:Fwooper - Found at any of the 4 Dens. West of Feldcroft:Giant Purple Toad - Found at any of the 8 Dens. North of North Ford Bog:Hippogriff - Found at any of the 3 Dens. 1x Male & 1x Female rewarded during main story quest "The High Keep." Forbidden Forest just North of Jackdaws Rest:Kneazle - Found at any of the 4 Dens. East of Irondale:Niffler - Found at any of the 6 Dens. South West of Jackdaw's Rest:Thestral - Found at any of the 2 Dens. Bred during the "Foal of the Dead" side quest. South of Coastal Mine:Rare:Graphorn - Found at the 1 Graphorn Den. Only 1x Graphorn spawns at a given time here. Must complete main story quest "San Bakar’s Trial" before they will appear at the den. South Clagmar Coast: Unicorn - Found at the 1 Unicorn Den. Only 1x Unicorn spawns at a given time here. Forbidden Forest: