I got a new to me bow last week. The manufacturers recommended brace height is 7 1/4 - 8 inches. How do I know when I find the optimum height?

When you find the right b.h. the bow will have the least amount of vibration, and will have the least noise, provided you have the correct arrow spine and weight.

when your arrow flight is as smooth as possible, the bow is quiet and you have achieved the maximum possible push on the arrow . . . or the combination of above that you can at least live with.  A small brace height gives the most impulse to the arrow but will be noisy.  High brace height robs you of some speed but is quieter and may make the bow feel harder to pull (you have to load it up in a shorter distance).

Kinda a chicken and egg thing, though, in that the right spined arrow will fly great and the bow is quiet at the optimum brace height, yet it's hard to get the proper arrow spine without having the brace height set first. I like to see what others who have the same bow are doing on their BH and use that as a starting point. Better yet, talk with the bowyer.

What bow is it Ermont?

Whenever I pick up a new to me bow I just figure that the brace height and nock point are WRONG. I set my nock 5/8 high and BH at 7 1/2, grab some arrows I know are going to be pretty "close" then head to the range. I shoot 3 or 4 arrows at a time, untwisting the string by 3 or 4 twists each group till my BH is an inch shorter. Then do the oppisite till my BH is an inch higher than where I started. By then I've got a "feel" for the bow and can then walk it back down till its pretty much right on. (sound/vibration) I then will bareshaft to tune bow/arrow together. I might have to "tweek" BH and nock point a little to get it right on but usually not much......YMMV

Cold Creek Black Feather.

Maybe a Crow Creek Black Feather...?

Make sure to take off the string silencers.
You will tell from the sound when you are at the correct BH for the bow.
Follow what hvyhitter said with no silencers.

Yeah I screwed up. Crow Creek!