Burstone Quarry is a fairly eventful level: lets you move fast but also requires you to be very precise about it, has a boss fight and a Shard, as well as triggers an invasion of the Ura after returning to the Bastion.

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Burstone Quarry, New Game

Route

NG/NG+ Any%/ASL

The first two gates can be either clipped through from their right side or, in NG+, destroyed with one Scrap Musket shot.

The majority of the level from here on out is very movement heavy. The way will be sometimes "blocked" by a Pincushion, but it is possible to get around them with some careful rolls (or destroyed with attacks, which is much slower). When you roll around Pincushions, make sure that you do not end a roll next to one or roll directly into one, as either will make it much more likely that you get knocked back by the Pincushion, causing you to fall. There are a lot of stones all over the level that can cycle their positions when you activate switches that are also spread around, but it is recommended to not use any switches at all - it is not hard at all to roll past all of the stones without activating any of them.

NG+ - some small parts of the level can be skipped with a short Pike Vault.

Once you have rolled past everything imaginable, you will reach the arena where you will have to defeat the boss of this level, Sir Lunky.

Lunky Drop

360 DROP IT

The fastest way to kill Sir Lunky is a quick-kill trick called the Lunky Drop. Basically, it involves manipulating Lunky into jumping off the level to his death.

When you enter the arena that Lunky lives in, immediately roll up to the top-left corner of the arena. Clear the offscreen debris next to Lunky with the Pike or Musket (make sure you are far enough away from Lunky to not auto-aim at him instead of the debris, or you won't have enough time to finish the trick). Walk to the top corner of the dark tile at the corner of the arena, and wait there. If you are positioned correctly, Lunky will jump off the arena to his death.

If Lunky does not fall off, he will likely be sitting at the edge of the platform. If you have the musket, you can shoot him to knock him off. This will lose only ~1 second. If you do not have the Musket, you must actually kill Lunky to progress.

HaosEdge's tutorial explains and shows this trick in detail.

Completing the Quarry

After killing Lunky, get over to the first gate at the bottom of this space as quickly as you can and stand in from of it. This will cause the end sequence for the level to trigger sooner. When the gates open, walk down, open Zulf's dialogue, close it right after opening, and then wait for the switch on the left to activate. Immediately use it. This will start raising the cage, letting you pick up the Shard when it's about halfway raised. You can now leave the level. Returning to the Bastion from Burstone Quarry will trigger the Ura Invasion.

Ancient Knowledge

It is possible to get to Lunky's arena with a pike vault or with a jaunt, but it doesn't accomplish anything because this way you won't hit the trigger that spawns Lunky, who you have to kill to be able to progress.

It is also possible to get through the 3 gates after Lunky with jaunting, but it is impossible to pick up the Shard or leave the level this way.