For a while during the Mists of Pandaria launch, the most profitable thing you could do with these green quality gems was to turn them into enchanting materials, however now this might feel like a lot of work for not a lot of money. For example: three green gems make a green BoE jewelery piece that will disenchant into 1-8 Spirit Dusts or 1-5 Mysterious Essences, averaging out to just over 5 dusts (assuming you're eventually trading dust up for essences). If the price of dust is 5g, that's pretty good. It's much more likely, though, that the price of dust on your realm will be something like 1.5g. It's not even worth disenchanting at that price -- that green will vendor for just under 9g. Even if the price of enchanting mats is high enough for you to choose to disenchant the greens you'll still have the logistical hassle of trying to manage inventory.
The blue quality gem cuts can only be obtained through daily research or using 3 Spirits of Harmony, which can usually be put to a much better use. This research is by color, so if you're looking to get the very popular Vivid Wild Jade cut, you might be unlucky and only get it on your 18th day of research. This may be one of the reasons that perfect cuts, which can be procced off trainer-learned cuts, were designed to be as good as blue cuts. You don't have to wait to get into that market, you can simply cut a stack of Alexandrites into an average of two Perfect Vivid Alexandrites. The 18 normal quality cuts it takes to make two perfects generally won't sell in anywhere near the volume you'll produce them in, but they vendor for 17g a stack.
Another part of the shuffle that perfect cuts can help in is meta transmutes. The orange, purple, and green blue quality gems it takes to transmute the meta Primal Diamonds are often worth more as part of a meta gem than they would be cut. Orange, purple, and green perfect cuts are a very good way to get this part of the market without sacrificing some meta gem production.
If you use an addon that integrates your crafting and auctioning (like TradeSkillMaster), you'll have trouble with perfect cuts. While you can manually search for a perfect cut to see what the expected price would be, it's a very manual and time-consuming process to search for every single possible cut you could make. Also, unless you have a good memory, you're going to have to keep notes on the ones worth crafting. Luckily for us, there's a website that does this for us!
If you go to the Undermine Journal, click on your realm unless you've already set that up, click the header "crafted" and then the subheader "Jewelcrafting", you will be sent to your realm's Jewelcrafting page. The part we're most interested right now is The Gems Array. Each of these tables represents the prices for a certain color of gem. For example, this image shows the uncut, perfect cut, and rare cut prices for all the ways you can cut a blue gem. If you hover over a price, the tooltip shows you the exact item you're seeing.
While this may tell you exactly what items are being posted for, it doesn't tell you what price they'll sell for. I doubt many people are willing to pay more for a perfect cut than a rare cut, so long as the stats are the same. Still, it gives you a nice way to know which cuts to avoid all together.
With this information, you'll be able to queue up the cuts you want in your addon (or just do them with the normal UI). If you use TSM, you'll want to ensure that the green cuts are enabled in your tradeskill settings, and once you have the perfect cuts in your bags, you'll have to separately and manually add them to your auctioning groups. All of the amazing integration that TSM has for non-procced cuts and crafts are, so far, not working for perfect procs. Still, you only have to add each perfect cut to a group once.
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