If Brown Turkey has a large Breba or 'first crop', I would wait until you've harvested that and then prune. Could be in June for you. If it is not a 'breba' or first crop fig, then you could start after the worst of winter is over, like most other fruit trees, in late February. Do not try to take too much wood in any one year, or you'll get a bunch of suckers from the base and more of a thicket.
Try to make mostly thinning cuts, but you may have to head back a few of the tallest leaders.
Don't 'flat top' the tree.

Here is a link that might be useful: Brown Turkey Fig care

The BT (and the Celeste) are the 2 most grown figs within
the whole USA. The Celeste is supposed to be a 'main crop'
fig (meaning fruit on fresh current wood). The BT is
considered as a 2 crop fig (meaning fruit on both last
year and current) wood.

For the sake of completeness, some other figs are known
as 'breba-only' e.g. the Dessert King
(fruits only on last year wood, fruit on the new wood
needs caprification).

Of course, there are those super 'other' figs that perform
extremely well in 'balmy-CA/equipped-with-the-fig-wasp).
e.g., the (in)famous Calimyrna (Syrna) fig.

Hi, I live in the North West, Columbia City Or., I have a very large Brown Turkey tree about 15-20 ft. tall, it is very health and produces lots of fruit every year. I would like to prune it down to a better size, So I can get the fruit before the birds do. The bottom branchs are 4-6 inch thick, with lots of open space, no crossing. If you could help me with, when and how to prun my beautiful Brown Turkey, with out making it look like a mess or damage, that would be great. Thank You! Oneata