For 1 Cocktail:
Spread the Basil salt onto a small plate, moisten the rim of a 6-ounce martini glass with the lime wedge, and upend the glass onto the mixture to crust the rim.
In the bottom of a cocktail shaker, muddle the basil leaves with a cocktail muddler or a wooden spoon until roughly mashed. Add the tequila, lime juice, Cointreau, Peach (or Mango) Puree, Basil Syrup and ice. Cover and shake vigorously until frothy and cold; tiny ice crystals will appear in the drink after about 15 seconds of shaking. Strain into the salt-crusted martini glass and serve immediately.
For a Pitcher:
In the bottom of a pitcher, muddle the basil leaves with a cocktail muddler or wooden spoon until coarsely mashed. Add the tequila, lime juice, Cointreau, Peach (or Mango) Puree and Basil Syrup. Cover and refrigerate until chilled, about 2 hours.
Crust the rims of 8 6-ounce martini glasses as described above. Fill a cocktail shaker ¾ full with ice, and pour in 1 ½ cups of the margarita mixture (be sure to include some of the muddled basil from the bottom of the pitcher). Shake, strain into 3 of the salt-crusted glasses, and repeat for the remaining margaritas.
Measure the sugar and ½ cup water into a small saucepan and set over medium heat. Add the basil and bring to a simmer. Cook for 2 minutes, remove from the heat, cool and strain. The syrup will keep for several weeks in the refrigerator, tightly covered. (Drizzle leftovers over halved, peeled peaches, roast at 450 degrees until tender and browned: these are incredible served with ice cream.)
Pull the basil leaves from the stems (you’ll have 2 loosely packed cups). On a rimmed baking sheet lined with a silicon mat or parchment paper, spread the leaves out in a single layer. Heat your oven to its lowest setting (160 degrees is ideal), then slide in the basil and heat it—really it’s dehydrating—until the leaves are dry, about 30 minutes depending on your temperature. (If you oven has a convection fan, use it here.) Cool. Using a small food processor, electric spice grinder or a mortar and pestle, coarsely pulverize the dried basil (you’ll get about 2 tablespoons), then stir into the coarse salt. Yield: About 6 tablespoons.