In a small saucepan set over medium-high heat, add the water and sugar and stir occasionally until dissolved. Pour the water-sugar mixture into a blender jar, add the vermouth and chopped mangos, and purée until smooth. Strain through a chinois, then pour the mango syrup into a bottle.

In a Collins glass, use a muddler to crush the cilantro and 1 sprig of the mint until they are roughly mashed and fragrant. Add the tequila, 1 ounce of mango syrup and crushed ice and stir to combine. Garnish with the remaining mint sprig.