What’s the best way to get rid of a bulging brow?

It is very hard to comment on your question without photos and/or in person evaluation. In most cases bulging is result of large frontal sinuses. Reduction and flattening of bone using the burr is very efficient way of correcting this deformity. All is done through a hairline incision or with endoscopic approach, using 3 small incisions hidden in the hair. In person consultation with an experienced plastic surgeon, familiar with this technique is recommended. Good luck.

The short answer is that sometimes bulging brows can be shaved down, but you should post some pics and ask your question again.  Usually a CT scan is necessary prior to considering surgery.

This is an operation I invented 40 years ago and have done more than 500 without a complication. Make sure you see a very experienced surgeon for this. 

Brow shaping is something we do here at my center 4-5 times each week.  Typically it is a large frontal sinus and can certainly not be treated with a bur. We typically set the bone back to restructure the forehead appearance.  Additionally, as a male, you would not want any incision in front of your hair.  We would do the incision high and far back on your head to prevent visible scarring.Watch our videos to learn more. The procedure takes about 1 1/2 hours total.

Brow bone reduction surgery reduces the size of the protruded brow bones. It's the best option to get rid of bulging brow bone permanently. 
Depending on your facial bone structure and individual beauty goals you could also combine Brow Bone reduction and Forehead fat graft to achieve perfect forehead line (it's also called Upper facial feminization surgery). 
An experienced facial bone plastic surgeon need to examine your case personally to give you the best recommendations. 
Best wishes, 
Park Sang Hoon, 

An in person evaluation well allow your provider to know if this is due to bone or soft tissue. If it is due to soft tissue you may see some softening of the area with Botox. Botox is a simple injection that can decrease muscle movement for around four months. Continuous usage can also decrease the size of the muscles. If the bulging is due to bone it may require a surgical procedure.

Some Facial plastic surgeons can shave down the brow to make it look less prominent.  Find such a doctor

The best way to permanently eliminate a bulging brow is through the Irregular Trichophytic Hairline Incision technique that I developed and described many years ago. Usually this problem is secondary to increased bone thickness which can be reduced with a diamond burr.