Pay whenever you can
Miles are not getting any easier to acquire, so I pay as often as possible. However, as UA 1k (and AA EXP with a nice pile of AA miles,) I'll use the miles for an award in the future and keep checking for price drops or other options to open up if I want a particular flight or time, as I can cancel the rewards w/o cost or fees.
Other considerations would include being close to a milestone for elite status/UPG's/Million miler status/promos or just accumulating miles for a better (aka Business class) award, as I don't like coach awards for international travel unless absolutely necessary. While elite status is generally not worth the cost and effort these days, there are some worthwhile benefits and may find it useful to hit that.
One other thing to look at is a one way award to Europe, as there are so many deals out of Germany, i.e. LH's $500 RT to many US cities. Sure, not easy to get to your home airport, but with some effort, you may find something that brings down the total cost, in both miles and cash. You can either plan another trip back to Europe, or just toss the return-as a German court recently made this OK to do, invalidating some of LH's T&C's.