To make the Simple Syrup: In a saucepan, bring water and sugar to boil, simmering until sugar is dissolved.
To make the Pear-Fennel Puree: Chop pear and fennel and combine with water and sugar in food processor. Process until the mixture is a thin puree. Add sugar if needed.
To craft the cocktail: Pour green Chartreuse into martini glass and swirl to coat glass. In a cocktail shaker with ice, combine pear vodka, simple syrup, pear-fennel puree and pear brandy.
Shake well and strain into coated martini glass. Garnish with a large fennel frond.