Fill six martini glasses with ice, then water; let stand until glasses are very cold, about 5 minutes (or chill them, empty, in the freezer). Meanwhile, for each drink, push 1 or 2 slices ripe white peach and 1 slice fresh hot chile (try a red or green jalapeño or a Thai chile) onto a small skewer or toothpick. Sprinkle peaches lightly with kosher salt. Empty glasses and pour 1 to 2 teaspoons dry vermouth into each. Swirl to coat glass, then pour off and discard most of the vermouth. Fill a 3-cup martini shaker with ice cubes and add 1 1/2 cups top-quality gin or vodka. Seal and shake until cold, 4 to 6 times. Strain into glasses. Garnish each with a peach-chile skewer.
Note: Nutritional analysis is per serving.