Hi! I don't know how to obtain essences for Ryori specifically, and I know that you can search for specific pets and nurture them to get that essence, but is there any way to be able to nurture one species quickly in one go like in the breeding grounds?

As you noted, there is no way to do a species specific search in the nurturing grounds- this is a deliberate choice on Krinadon's part. An option is that if you can find a player with a lot of egglings/hatchlings of the species you want to nurture in AS you could potentially search that user in the nurture grounds (though obviously they may also have other species.)

Thanks!

While it likely wouldn't be quick, you could use Advanced Search to find Ryos to nurture.
In the Search Pets section, you could set the Species to Ryori, and the Maturity to Eggling and Hatchling, and then click search.
That should show you all the current Ryories that can be nurtured, and while it would be slower(Since you'd have to open the page for each pet), you would be able to specifically nurture Ryori.

However, going with Scathreoite's suggestion would likely be quicker and more efficient since you're looking to nurture quickly. ^^