Don’t name a cocktail after a woman—that's the lesson here. This is an elegant aperitif cocktail that is often made with too much thought.
Often referred to as a "Vesper Martini," this is an unfortunate portmanteau tantamount to saying "kitty cat." There is a Vesper cocktail and there is a Martini, (that has fewer acceptable modifiers than you have fingers) but do not say "Vesper Martini."
The Vesper is highly branded and calls for Gordon's and Lillet— why?
This seminal cocktail from Casino Royale is casually ordered as if that is an acceptable way to create a cocktail. It might be better to inquire about rather than demand a brand, so consider how you set the tone. To address brands, Gordon’s is still above average but is amazing for the cost. In the 50’s it was 94 proof, so choose a higher proof when picking a gin now. As to “Kina Lillet,” Lillet has come out over the past few years and said that they have never changed the recipe. Brands change even if recipes don’t. Proof, extractions and especially grapes in wine (like Lillet) change. Is it lighter now? We’ll never know. However, Lillet is within a fortified wine the category called “Americano,” if you want a more firm fortified wine, try Cocchi or a half dozen others. As to the Vodka, no brand is named because Vodka would have been exotic at the time. Perhaps it would have been like calling for an armagnac brand in a modern bar.
Stir or shake, doesn’t matter. Most bartenders shake a drink like they are about to fall asleep or stir a drink like a meth addict trying to stab a grape with a spoon— that is to say, poorly. Shaking is about adding air and effervescence to a cocktail, stirring is about keeping a silky texture and both dilute about the same.
Shake hard, or stir very gently and long enough to explain the plot of your favorite movie. Chill the cocktail glass before pouring the drink. Zest the lemon over the cocktail. You’ll be fine.