Down south I wait till fawns are at least 5 months old or after Thanksgiving. Most years I wait till after the first of Jan to take any does I need to take. If I am thinning numbers to tighten the ratio and slow reproduction I am targeting the older twinning machines out first. Fighting a numbers battle is easier with less fawns on the ground. The does without fawns will generally be the 1.5 yr or 2.5 yr old does, which should be better genetics than the older does. Does with one single fawn or no fawns will generally be in better shape than does with twins. They will have a head start on next year once they are bred. I really do not worry to much about killing any does bred to the bigger bucks since a mature buck breeds very few does per year. If you like the way your bucks look top to bottom in each age class then all the does will be bred to good bucks. I like a 45% fawn crop every year, which means that if I am seeing a lot of does without fawns that I have a lot of 1.5 yr old does that year. As long as a doe has raised one offspring I am fine with taking her out of the herd. The younger the herd the better off you are in the long run.
For those worried about shooting a doe with fawns, I would suggest if you can sex the fawns then shoot the doe fawn(s) also. It is a number, it is a mouth and it is a doe. I read an article one time that stated that if you shot a doe and she had a buck fawn, the orphaned buck fawn would generally stay in that area for longer than if the doe was alive.

We typically keep the pasture quiet early season to keep bucks hanging around and shoot does late. By the end of the season fawns are ok on their own and we just shoot whatever makes itself available. I love to shoot a does when its the only deer there. Bad for judging size etc. but good for not educating any more of their buddies.

While it may be true an old doe is quicker to catch onto you, she's also more likely to detect natural predators that would otherwise have an easier time taking down fawns and less experienced deer.

Well, I go against the grain, I try to shoot the mature does in order to get/keep the buck/doe ratio as low as possible. a yearling doe will have a singe fawn and about 73% of the time it will be a buck.

I have just recently, last 2 years, started taking doe. I only take ones that do not seem to have a fawn.