If you are doing lots of buttons you want to make the mount on off fast.
making a 2 pin drive mandrel that fits in your 4 jaw chuck or better yet on a wooden Morse taper will be well worth the time.
the two pins fit in opposite button holes.
drill the holes first. either make a template or a jig on the drill press easy to do with square pieces.
This is a mind numbing job so remain alert or you'll get hurt.
put brass rods in two holes to stack the buttons 10 to 20 deep. Mount the stack between centers centered on the holes.
Cut the outside diameter. If you aren't good at turning cylinders, you will be. In the beginning you could cut 3 or 4 pieces to diameter very carefully an insert them in your stack to use as a guide. After the first stack the two end button will have center marks an you probably just reuse these. I would use a cup drive center or steb center.
turn all the faces
then make a wooden jamb chuck with a hole in the bottom.
press the blank in the jamb chuck turn (and sand if needed) the face.
pop out with a knock out rod.
make a wooden mandrel with two pins (brass rod sections) that fit opposing button holes. on the tailstock you'll need a small diameter flat 1/2 or less covered with leather. You can turn a wooden cover for most any live center that has straight sides on the revolving part. If you have a oneway live center get a Flat extension point from Bruce Campbell
http://www.artisansworkbench.com/Product info/Extension Points/extension_points.htm
glue on a piece of leather with CA or something you can clean off.
put the button on the mandrel bring up the covered flat live center, turn the edge. This has to be done last as you may change the diameter ever so slightly. sand if need
have fun.
Al