At least I think it’s an Italian liquer - it was served to me in an Italian restaurant. It’s an after-dinner thing intended to help settle your stomach when you’ve overloaded your plate a bit. 
I vaguely remember being told it was some sort of bitters. It didn’t have a great taste, but it was nice and warm on the way down, and it really felt like it burned away all the food that was sitting like a lump in my stomach.
Could it have been Angostura bitters? Any other possibilities?
Was the drink clear? If it was, I would say you had some Grappa. It is a drink made from grape seeds and stems, IIRC. The stuff that I tried was so bitter that it was undrinkable.
Maybe Campari? Was it red? Although Campari is usually drunk before dinner, not after.
It’s been a long time since I had it, but if I remember correctly it was brown, & served in a fancy shot glass (so no ice).
Fernet Branca, I’ll bet.
liquer, I hardly know her!
cher3, I think you may be on to something here. I’ll have to arrange a special field test at Valentino’s next weekend.
ReillyDog: Har! 
Perhaps it was Galiano?
Sounds like grappa to me! Unlike Adam Yax, I think the stuff is pretty good. And yeah, it is made of the remains of grapes after they have been pressed for wine.
Grappa would certainly cause a nice warm feeling, but it’s clear, not brown. (You can see some on the link in my last post.)
And Galliano is pale yellowish and very sweet.
whoops! skimmed over the post where the drink was described as brown…
I’m starting to loose focus this late in the afternoon
Vin Santo is amberish in hue.
Amaretto is brown. It has a strong taste of almonds.
Is it necessarily Italian? Underberg Bitters fits the description in the OP, being brown and a herbal digestif, but the drink is German.
I’ll admit that I have no idea what adult beverage o.p. is referring to, unless it is grappa. No intended hijack, but does anyone have a link to an “encyclopedia of world-wide liquors”? In my 24yr USN career I’ve bumped into some great (and truly hideous) liquids but haven’t found a compilation of everything that’s out there.
Sounds like Tuaca to me.
If you saw the bottle, maybe this link will jog your memory.
I keep a bottle around most of the time. It is commonly served neat after dinner as well.
Strega, perhaps? Did it have a taste like spiced anisette?
Busy here since I left yesterday! I should ask about liquor more often. 
Re the latest questions -
I didn’t see the bottle, since the shot was brought out in the glass. It was definitely not Amaretto. I don’t think it was Strega or Tuaca, either - it was not so much spicy as it was medicine-y, very bitter. What does Vin Santo taste like?
Nostradamus, I suppose it could have been a German liquer. I’m only assuming it was Italian because the restaurant had a definite southern Italian bent.
’Uigi, I tried searching for some sort of liquor database but got only lists of distributors. Maybe someone else will have better luck.
Fillet, personally I think it was Fernet Branca, like cher3 said, but what about calling the restaurant? The bartender would probably know right off what you’re talking about.
Unless it really was Angostura bitters, which says or used to say right on the bottle that it was good for an upset or too-full tummy. Although, I always mixed it with soda or 7-up instead of drinking it straight. It makes a nice refreshing drink.
I’ve never seen a case of hiccups that couldn’t be cured by:
Biting into a wedge of lemon that’s had a couple drops of Angostura bitters applied to it.
missbunny, if I don’t get the chance to run a field test this weekend I certainly will call (but the field test would be more fun ).
Thanks, everyone!