vodka + vermouth, but not a "vodka martini"
What is the objective? Usually people ask about ingredients because they like their flavors. But since vodka has little flavor (essentially none when mixed in a cocktail), vodka + vermouth tastes like boozy vermouth. What you add to it will dramatically change its flavor.
For example, equal parts vodka, vermouth, and Campari would be a dry, conifer-free version of a Negroni. Cynar would be nice too. The vodka would serve only to raise the proof.
Another idea. Make a more grown-up Cosmo. I haven't tried this, but something like:
1.5 oz vodka
0.75 oz dry vermouth
0.5 oz lemoncello
0.25 oz unsweetened cranberry juice (not cocktail)
expressed lemon peel
You'd have to play with the ratios. Cranberry juice is very sour and a bit bitter. The lemoncello will sweeten it, but the vermouth will dry it out. The idea would be to have a faint pink color, a subtle, bitter cranberry flavor, and an balanced (not sweet) cocktail with a wine base.
Never had this combination although I rather enjoyed the Club Cocktail I had (equal parts vodka and white port, 1-2 dash Angostura). Worked great with Bison Grass Vodka for the cinnamon spice in the spirit complemented the Ango well.
I can understand a dirty vodka martini for it brine-flavored booze. Vodka + vermouth is just watered down vermouth, er diluted down vermouth.