I think that it would be faster and easier to learn to knit in the round for these. Knitting in the round is great to know, lots of things are faster and easier in the round. That said, there are owl cable patterns for flat knitting so you could try incorporating one of them into flat knit mitts. I strongly suggest that you try knitting in the round.
Denial, anger, bargaining, depression, acceptance...the five stages of frogging.
You can knit them on straight needles and add a stitch to each edge to seam them. The cable crosses would be done on the RS rows and for the WS rows work the sts as they appear - knit the knits and purl the purls. Probably.... You could start one and work a few rows past the wrist ribbing and see how it goes.