The noise is coming from your handlebars, but in my experience noises can be deceptive. It may be your stem-handlebar interface, but it may not be isolated to that one spot.
The first thing to check is that the tabs on the face plate are not meeting the body of the stem when everything is torqued down. If you can see a gap there, you should be ok. If they are touching, however, the final torque will do nothing to hold the bar in place. This is sometimes a problem when new handlebars/stems are installed with parts that are spec'd a few millimeters off (i.e. in the example stem you provided, the bar clamp is 25.4 mm. If one were to replace it with a road stem with a 26.0 mm clamp, it may fit well enough to hold the bar without translating all of the tightening torque to the clamp area.)
Another component to investigate is the headset. Lots of headsets are press-fit items and if the fit is poor, the races will make noise. This would be noticeable if you were to hold the front wheel with your knees and pull the bars side to side. It may not be apparent from examining with everything in tact, so some amount of disassembly may be required. A sign of a poor headset fit is fretting marks on the head tube. You can find Loctite or similar products designed for press-fit applications.
Another headset issue that may be causing your clicking is indexing, or brinneling. This occurs when the ball bearings, usually in the lower race, wear evenly spaced dents into the bearing surface of the lower frame cup. This is more noticed in steering, but if the headset is over tightened enough, it may 'click' into place.
If the issue persists, check your front hub to make sure the cones haven't become loose. This would allow the wheel to move slightly side-to-side making a noticeable tick. If you have a suspension fork, check to see that the stanchions haven't become loose in any way.
To further muddy the water, you may want to check the frame very thoroughly for cracks or dents, especially around the headtube welds. This may not cause a noticeable click all the time, but a visual inspection will save you lots of headache if you find it early. If the frame is anything but steel, consider trashing it. Some frame materials have a tendency to fail catastrophically and unexpectedly.
If all else fails and your certain the clicking is coming from your handlebars/stem, replace them. They're expendable items with a definite lifespan. A failed handlebar or stem will almost always cause a serious crash.
Good luck!
