|I bought a shirt that has a cool treatment for those turned-up cuffs that have a slit, usually found on three-quarter sleeve shirts. (It is made by Sportif, whose sportswear is high-quality, sewn to withstand a nuclear blast.) It was done this way:
Shorten the sleeves to about 1 inch beyond the point where you'd want to turn the cuff, and set them in to the body flat. Sew up the sides of the shirt. The Sportif shirt used fake flat-fell seams, but you can do it any way you want.
Make two separate cuffs with rounded ends like a regular shirt cuff: two layers of fabric (maybe including a light interfacing), sewn together, leaving the sleeve side un-sewn. The finished cuff should be the the same width as the sleeve plus about an inch (or exactly the same width as the sleeve, if you don't want an overlap). Turn and press. Topstitch 1/4 inch along the finished edge.
The sleeve/cuff seam is a French seam. Put the cuff on the inside of the sleeve, so the cuff is against the wrong side of the sleeve fabric, and placed where you want it. Typically, the slit is about a quarter of the sleeve circumference from the sleeve seam around the back. If you're overlapping the cuff, make sure it overlaps in the opposite direction to the cuff on the other sleeve. Sew a 1/4 inch seam. Press and turn so that the cuff is sitting against the right side of the sleeve. Press again. Sew a 5/8 inch seam. Press and turn the seam allowance to the sleeve side. Topstitch 1/4 inch from the seam.
You need a sewing machine with a free arm that's small enough for the sleeve to go around. This treatment solves the problem I have of slits that split and ravel as soon as I start to wear the shirt, no matter how carefully I do the slit.