Surgery to remove hair?  Never heard of that, but there is laser hair removal or electrolysis

If your lucky HRT will slow you beard growth and maybe thin it out a bit making it possible to get a little more mileage from shaving.As Caysee said, laser and/or electrolysis are the only ways to remove it completely.

Even if there was any surgical options, I'm sure possible scarring and nerve damage would negate any benefits over laser/electrolysis.

Been doing laser since august 2012, and HRT since may 2013. Even though there is little hair now, it always has a slighty darker shade on the upper lip. Like if dormant, non activated follicles still were there.

I'll be starting with electro In may. The last survivors actually keep bothering me..,

I did 2 years of laser on my face and neck. Having dark hair and light skin made me a prime candidate for laser. Light hair or dark skin and electrolysis seems better.

Everything I've read suggests that HRT won't affect facial hair all that much if at all. I've not yet heard of surgery.

Before I started laser, I had a makeup lesson at a boutique that catered to cross-dressers (as I identified at that point) and trans women. There I learned how to use makeup to hide the beard shadow. It was better than nothing, but I'm really happy with my laser results. I rarely use makeup these days.

I've been having electrolysis sessions off and on for the past few years. Lately, I've been going as much as twice a week (worth every damn penny). Most of my face looks like there has never been a hair on it.  Been on HRT for years also, and my doctor always says Spironolactone is good for hindering facial hair growth. My neck and cheeks literally feel as smooth as glass. I've still got a few more sessions "under the needle", so I'll say a big no to HRT doing it alone. I used to be able to grow a full beard at one point, as horrible as that sounds.

Laser Hair removal, takes about 10 - 15 sessions.  Painful and redness for a couple of days after treatment.  Electrolysis, similar number of treatments, also painful and leaves a rash for a few days after each treatment.

Hormones do not get rid of facial hair.

HRT has slowed facial hair growth but only a tiny amount. It slowed body hair growth a fair bit.

As others have said you will need to find an actual hair removal solution to remove that shadow.