I have sought confirmation for Elixir about the name, ingredients, history, and attribution for this cocktail. This is obviously not a relative of a true Martini and perhaps has a better name.

Sounds similar to the Dreamy Dorini Smoking Martini created by Audrey Saunders that I saw mentioned on eGullet:
http://forums.egullet.org/index.php?/topic/82754-simple-vodka-cocktail-…

H. Joseph Ehrmann kindly wrote back with the original recipe, notes, and an aritcle from November 30, 2008 in the New York Times where it was featured. I have updated the recipe's name, history, date, and reference to reflect these source references.

Thanks, Dan, for doing that, and thanks to H. Joseph Ehrmann.

This is a perfectly balanced drink. I've been making it with Eagle Rare 10, though will be trying with Weller 107 -- I'm curious to see how it performs with a higher-proof, wheated spirit.

I've made this now with both bourbon and rye. It's an excellent drink in both iterations: spicy, dark, strong, not too sweet (though maybe still too sweet for some), integrated and balanced. The rye version wins by a hair for me as it makes a dryer drink and its spice plays well with the bitters. (not having celery bitters, I went in a different direction and used Dutch's Colonial, which have a woodsy and anise-y character that plays well with Cynar and rye. Not sure whether that changes the cocktail enough to merit a new name.) The rye and Cynar open the sip while the Maraschino lingers in the finish. The rye draws some cool dark cherry notes out of the Cynar and the two together develop an almost cola taste, but an herbal, 19th Century cola. Moderately but pleasurably bitter. Recommended for fans of the New York-themed Bitter/Brown/Stirred standbys (Little Italy, Red Hook, etc). My only complaint is its name: a great drink deserves a better one. 4.5/5

A little too sweet. 

See also the Pink Lady, which includes egg white.

Curated to add credit for this adaptation, per reference link. Thanks @dustinw84

I keep coming back to this one as we transition into fall. So good! I cut the grenadine to 1/2 and I think it's improved, for my taste.

Solid drink. I would cut back the sweet vermouth to an ounce, though.

.25 oz lime made this more to my taste. I didn't have the suggested rum, used Rhum Barbancourt 8 yr.

Lime would definitely help; it needs acid and tartness. All I can really taste is the Chartreuse; the Averna is basically buried. I give it 4/5 mainly because I love Chartreuse to death.

I agree. I tried it as written. Blech. Too harsh. I added the 1/4 oz lime. Mmmmh marvelous. And it was so little citrus it just elevated it while still leaving it to feel like a spirits driven drink.

This works really well with a liqueur made from bay leaves or seeds in place of the Charteuse, if you can find (or make) one.
Green Chartreuse may be too strong - try yellow?

I too did a lime-tartness thing, but I didn't want to add too much to it, so I used lime bitters and acid phosphate. Pretty good.

Several references online quote Chris Hannah as saying his preferred rum for this is Plantation OFTD, which would make for quite a different drink than with the Cruzan. Furthermore, he calls for Amer Picon, but says that Averna makes a good substitute. To confuse it more, some references have the amounts for Chartreuse and Picon/Averna flipped. 

A few of Chris Hannah's recipe have drifted over the years such as the Rebennack. This would not have been invented with OFTD since it had not been invented. This is the one I sourced from Under the Table: A Dorothy Parker Cocktail Guide and wrote about here: https://cocktailvirgin.blogspot.com/2015/06/bywater-cocktail.html

1 3/4 oz Aged Rum, 3/4 oz Picon, 1/2 oz Green Chartreuse, 1 tsp Falernum, 2 dash Peychaud's Bitters, 2 dash Orange Bitters. Stir with ice and strain into a cocktail glass. Garnish with a cherry.

Added the 1/4 oz lime juice - *chef's kiss*.

For the record, Punch has the ingredients as:

• 1 3/4 oz aged rum
• 3/4 oz Averna
• 1/2 oz Green Chartreuse
• 1/4 oz Velvet Falernum
• 2 dashes each Peychaud's and Regans'

Express and discard orange twist; garnish with cherry. 

I have written the restaurant in the hopes of getting an authentic recipe. I did make this with, uh, liberal substitutions -- Punt e Mes for the Carpano Antica and Amaro Nardini for Creme de Cacao. I enjoyed the cocktail, although I would reduce the Mezcal to keep the smoke in balance, and I can see that more chocolate would be good. Still, a complex and entertaining drink, even with my approximation.

I've actually gone and asked specifically about this recipe.

1 oz. Mezcal, 1/2 oz. Applejack, 1/2 oz. Carpano Antica, 1/4 oz. Creme de Cacao, Barspoon Fernet Branca, dash of Xocolate Mole

Made this using the recipe from Mayahuel, as quoted in the comment above, and really liked it. Will make again. 

I curated this to conform with the newly cited link, which seems like it's closer to the actual recipe than the one we had.  Thanks,  Zachary

REALLY good. Was tempted to dock half a star due to labor intensity, but couldn't do it. ✯✯✯✯✯

I agree. A bit fussy on the front end, but distinctive and complete when all is said and done. I taste the booze and all the other notes that wrap it like a beautiful Pucci scarf.