As soon as you complete his Lure of Obsidian questline, Magnus will give you this three-parter. It is likely that you'll be able to line up at least Part 1 of this quest with Giovanni's quest of the same name (yup, Braving the Basilisk...confusing, I know!). Happily, they share some elements, so you can knock out multiple tasks with less effort.
Even after you complete Giovanni's eponymous quest, he'll move you along to another quest with similarities to this one, called Toy Time. The Basilisk Scales, Brass Gears and Toy Dragons will crossover with parts 1 and 2 of this quest.
Once you complete all three parts, Magnus will give you a new quest, a four-parter called Dyed in the Wool.
Part 1 of 3
Part 2 of 3
Part 3 of 3
Gather 4 Basilisk Scales
Craft 4 Basilisk Necklace
Weave 4 Basilisk Bandages