Sometimes called an afterthought thumb, I first read about it in Elizabeth Zimmermann's Knitter's Almanac.
This method is used in fetching but it is also very common in a lot of stranded, and Norwegian style type mittens.
I really like this method because it's seamless; sometimes when patterns have you place stitches on a holder and then cast on the top stitches you can get an odd seam right in the crease where your thumb meets the hand, and that can be uncomfortable.
The only trouble that some people may have with this method is that you cannot immediately try on your mitten in progress, as you can with mittens where the stitches are put on holders. But if you want to you can just knit a few more rows and then pick up the stitches instead of waiting until the mitten is done, threading the waste yarn through them so you can try on your mittens as you knit.
(see the working yarn, still on the right-hand side of the work?)
(Fetching uses a ribbed pattern. When knitting with the waste yarn, just knit plainly, and when knitting back across the waste yarn, knit plainly again, then pick up any stitch patterns again once you've passed the thumb. If using colour patterns, they can be resumed right away as you are knitting over the waste yarn. It is just easier to pick up the stitches later if they are all knit.)
See the purple bumps in-between the blue bumps? Those are the stitches you'll be picking up.
Keep a loop of the circular cord at the end, to separate the two sides.
The top row will have one more stitch than was originally knit onto the waste yarn, and the bottom should have as many as was originally knit.
(The fetching are open at the top, so I stuck them out the top, if making a closed top mitten, just send them out the wrist.)
Ta da! You are almost ready to start knitting your thumb!
If things are still a bit loose at the sides of the thumbhole, just pick up one more stitch at each end and knit it together with the next stitch on each needle on your first round.