For this method, you will bind your sleeve (or any cylindrical edge really - it doesn't have to be a sleeve) in the round, which will leave no bulky, scratchy seam allowances exposed.
The pattern pieces you will need are the sleeve and sleeve binding strip. If your pattern doesn't include a binding strip - fear not! This is super easy to draft, as it is a simple rectangle. The long edge should be the finished circumference of the sleeve (after any pleating or gathering is done), and the short edge is the width that you would like your binding to be, times 2. Add seam allowances to all four sides and you are finished. Prepare your sleeve by putting the ease/gathering stitched in the sleeve cap, and by doing any pleating or gathering or tucking or whatever is to be done to the lower edge of the sleeve. Basting your pleats or gathers in place helps this go more smoothly. I also find it helpful to press up one seam allowance of the binding strip (Sorry no picture!). Just put the crease in place so that it is easier to fold under when you get ready to stitch the inside edge down.
Next you'll sew the underarm seams of the sleeve and binding. Finish the sleeve seam in whatever manner you have been sewing the dress. French seams are nice. I simply serged mine. For the binding, you want this seam as flat as possible. Since it will be fully self-enclosed, there is no need to finish it. I sew a simple straight seam and press it open. Fold over your pressed edge and crease it again to even out any irregularity at the seamline.
Now turn your binding right side out and tuck it into the inside out sleeve, with the raw edges together and the seams lined up. They should be right sides together at this point. I place a few pins around the circumference to keep the layers from slipping as I stitch.
Now you stitch the circle. This actually does fit over my free arm, but I feel like I have better control when I can see the entire circle as I stitch, so I keep the entire sleeve above the bed of the machine. You want the sleeve against the feed dogs, just in case there is any sleeve fullness that needs to ease in . Keep the fabrics relatively taught as you sew, and the feed dogs will do the work for you. Sew all the way around your sleeve with a simple straight stitch. This seam will be encased in the binding, so there is no need to finish it, although you can trim it down if you like, or if your binding is narrower than your seam allowance.
Now press the seam allowance down toward the binding.
Fold the binding in half over the seam allowance. You want the edge you pressed earlier to just barely cover the seamline.
I stitch a few pins across the binding to keep it in place. Don't press the binding edge yet - wait until after it is sewn down. I used a slipstitch to invisibly hand sew the binding down on the inside, but if you would rather have a topstitched finish, I recommend glue basting down the inside edge and topstitching from the right side. (Sneaky tip: You can use regular glue sticks for this - you don't have to get the special fabric glue. Of course, test it on your fabric before you use it to be sure.)
Now give the sewn down binding a press to crease the edge and you are done! Your sleeves are ready to set into your garment. Here you can see how they look, inside and out. Hard to tell the difference, eh?
I hope you all find this helpful, and if you have any tips or tricks that help you, please share them in the comments!!