re: Does anyone know what brand vermouth Ruth Chris uses for the dirty goose martini?

Maybe they use vermouth (never had it) but usually a dirty vodka martini subs out the vermouth for olive brine. Of course to get the exact flavor profile you’d have to know what kind of olive brine they use. There are many commercially available dirty martini brines available, so best advice is to start there or just use the brine in a jar of olives.

If they do use vermouth it’s likely just a dash of martini and Rossi dry vermouth.

ETA: Wife loves them. House ratio is 1 part olive juice (use a brand called dirty comes in an apple sauce style pouch) to 3 parts vodka. Also you can make a big batch and pour back into an empty bottle. Stash it in the freezer. That gets you a martini (either gin or dirty vodka) that is on the border of frozen. Just make sure you add some water to the batch.