FWIW, this is from an old post on the 1911 Forum, when someone wanted to mirror shine the flats on a stainless Kimber... I saved it in case I wanted to try it sometime. I never did, so I can't say how this would work...
--------------------------------------------------
Polishing the Flats
Get some sand paper and emory paper. Local hardware store, auto parts store or Wally World. You will need a grit starting at about 250, then 400, 600, 800, 1000 and 2000, if you want for a real nice finish.
Get a piece of glass. I got one about one foot square and 1/4 inch thick. This is so you can tape the sand paper down to it.
Tape the 250 paper to the glass. Place it on a flat supported surface. Work the slide back and forth on the paper. Not round and round. Then move to the next finer paper and repeat. When you get through with the 1000 grit paper, you should have a mirror finish, almost. (Added this from a follow-up post) Go to 2000 grit sandpaper and use Flitz to get the final polish.
Polishing the flats is something you could do for yourself with little trouble. Use crokus cloth wrapped around a flat block. Start with a medium to medium fine and end with a very fine grit. Finish up by using an eraser and then some red rouge on a cloth.
Make sure you clean everything well with a good degreaser after you're done.
-----------------------------------------------