once your sleeve is sewn together, make sure you have a few points marked: where the cap crosses the shoulder seam, the two ease points and the point where the sleeve crosses the side seam.
now grab some polar fleece (which just about everyone has lying around) and cut two 2" x 12" bias strips.
mark the center of the strip with a notch and line it up with the top of the sleeve cap. then stitching at 4/8" (assuming a 5/8" seam) pull the fleece quite tight as you stitch it on. with a heavy coating, you will really need to tug at the bias strip; less so for something lighter weight. continue pulling and and stretching the fleece until you get to the ease mark, then you can release to light pulling until you get to the end of the fleece.
turn the sleeve around and do the same thing from the other direction. it's important that you start your stitching at the top of the sleeve, even though it's a bit awkward. if your machine doesn't have seam allowances marked on the left side of the needle, just put a piece of tape on the machine as a guide.
turn your sleeves right side out and make sure you know which sleeve is which (right or left). mark it if you have to, the last thing you want to do is try to attach the sleeves to the wrong side.
turn the coat inside out and place the sleeve inside the armhole. you'll see that it fits in with no further easing necessary, no pulling of gathering stitches, easy peasy. pin it in place, matching all your notches and dots and whatnot. you can baste the sleeve in place first if you want, or just stitch if you're feeling confident.
once your sleeve is in, round off the ends of the bias strip. lightly and carefully press the whole seam allowance toward the sleeve just with the tip of your iron. you don't want to get too far into the sleeve or you may get some unwanted creases.
now, step back, breath a sigh of relief and admire your handiwork!
next up... how to make shoulder pads!