The simpliest manner to do it is to tat a front element, with a nice center, that a back side and capture the cab /stone in between by sewing those 2 layers together or joining them by another round of tatting.
The method I found in a Russian forum (the author was Nina Prokoviewich but I think the method belong to Ekaterina Stepnaya) is, in my eyes nicer.
Irina Astratenko has also another method: 1 shuttle, alternating rings with seed beads in between. (work ring, turn work, place bead, work ring, turn, place bead, work ring, join... )
The beads are optional.
To understand the principle: you will tat both sides at the same time, by making some sort of frame.
You need 2 shuttles, because you will want that the threads you use for the rings end at the same point, to you can start the chain. Of course, you can use a shuttle and a ball, but you will need to turn and flip work.
You can choose any type of cabochon. My first try was a rivoli, 5 mm thick. I hesitates a lot to publish this method because I was thinking I am infringing the rights of Marie McCurry, but I am not. By the way: you should pay a visit to her page. Lots of goodies.
So there you go.
The last piece was a mother of pearl pendant:
You start by planning your tatting by copying the shape of your cab and then drawing. Use axes, just draw, imagine how it would be. 1:1 scale is very good. Working with cabochons and designing is not for beginers, you need to know your thread, in order to estimate the ds and to place the joins. Gauge your thread by making rings and chains increasing the ds systematicly. It can help you to plan your tatting. Like this you will have more or less your rough pattern.
I used for my test the Penny 30. All my rings were 6 - 8 - 8, all my chains were 8 bdp 8 (bdp: beaded diamond picot).
I choose the pink shuttle thread to work the upper rings and the blue shuttle to work the lower ones. The chains: pink shuttle thread was the core thread. Like this I avoided the thread to make unwanted switches.
I always start like this. Never a knot. I hide the ends with the first ds.
Work the first ring with the pink shuttle, do not turn. Take the blue shuttle and work very close the lower ring. Close and turn work. Hold both rings tight and start chain.
Snug the chain and prepare to work the next set of 2 rings. You will join the upper rings to form 1 round and the lower ones the same.
You will notice soon that they for the cage for the cab.
Continue until you will consider necessary for your stone. Make tests from time to time, to estimate how much you need to work. It is better to have it tighter, like this the lace will hold the stone very well in place. You will finish every round normally.
You will close first the upper rings. Then you will place your cabochon, hold it in place and work the lower ring, make the last joins and you will see already the stone caged in your lace. Pretty uncomfortable this last ring, but possible. Tat the last chain to finish your work.
Fot the white and lila piece, I worked an extra round.
I worked another piece with some beads. Not necessarily my thing, but I can show you how you can play after:
You can decorate this base as you like.
I would suggest you to start with a round piece and no beads (except the outer round).
Pay attention when you start any ring. The smallest bare thread between the 2 rings can spoil a lot from your work.
For a thicher cabochon, just make bigger rings. But still do not plan more ds between the joins because they will tend to go up, and not lay on the stone.