You can use a 4-prong cultivator and a steel rake to pull most sticks and twigs out of unfinished compost, if you don't mind a few chips being left.
Once the c9ompost has been rained on a few times, the fine particles will settle and leave the coarse bits on top. Then, you can either rake those off, or call them "mulch".
But I like sifting. If something doesn't go RIGHT through 1/2" mesh, I demote it or "hold it back a year".
I used to figure that slow pieces were probably full of microbes that break down slow pieces. Thus I thought that each piece I demoted to the next pile, or the "young end" of a compost row inoculated the younger compost.
However, since then I've read that the big pieces tend to be broken down by larger soil organisms, and THEN microbes can process them faster. If so, any "inoculation" probably requires wholesale transfer of the outer layer to carry along insects or whatever.
I still think that transferring some of the core of an active pile does jump-start a fresh pile with a big dose of rapidly growing microbes.
I can't find it now, but I've read about "phases" of composting, including (something like) these phases:
- raw material
- hot phase
- warm phase
- finishing
Different populations predominate in each phase, and then mostly get eaten by the population in the next phase!
so microbes can take it the rest of the way.