I worked the pick up round with and extra needle, it was a bit fiddly-clunky but ok. For every stitch I made sure the yarn passed around the pick up needle - forming a loop that I could later use to knit with.
Here is a short video showing the process, it might make less sense if you carry the yarn in the right hand - but this worked for me. The goal was to create a stitch with a yarn over over the second needle for every stitch knit.
After the pick up row I tucked the extra needle inside the mitt and pulled the tips out through the knit fabric. This stopped them from waving around and getting tangled in my working yarn and needles,
When the inside facing was the same length as the outside hem I transferred the two sets of stitches to one needle. One for one, in turn from the inside and outside needle, some people like to knit stitches off both needles but I find it less fiddly if I transfer them before I knit.