In our zone any time except early/late September through about late January.

I would hold heavy pruning that needs a saw for when it warms up in February.
Pruning shear and lopper cuts can be made any time February to September but August is a good time if your desire is to discourage regrowth.

It sounds like you're needing to make major heading cuts.
I would make those in February-March. Then you will get a lot of "water sprout" regrowth (I'm sure you've seen what I'm talking about at the ends of large branches cut off by the power companies). Use that regrowth to select and train your replacements out of.
You can thin some through the summer but save the majority of the thinning for August and you'll slow the regrowth for the following year. The tree will put it's energy into growing all those "water sprouts" which you will remove before they can give that energy back.
Or if you're after a faster branch replacement, never let anything but the one you've picked for the replacement to grow and the tree will have no choice but to put its energy into that branch.

Also heading cuts, where you cut a branch back to a point within the branch, tend to cause regrowth.
Thinning cuts, where you cut the branch completely off tends to not cause as much regrowth.