If the peaches are very sweet, add just three-quarters of the syrup to the puree.
Put sugar and 1 1/2 cups water into a medium saucepan over medium heat. Bring to a simmer, stirring until sugar has dissolved. Add zest. Simmer 5 minutes. Remove from heat; add lemon verbena. Cover; let stand 20 minutes. Pour through a fine sieve into a bowl; discard solids. Let syrup cool completely.
Meanwhile, coarsely chop peaches, and place in a large bowl. Crush with your hands or a potato masher, then pass through a medium-mesh sieve into a bowl; discard solids (you should have at least 1 1/2 cups puree). Add cooled syrup and vodka or grappa to peach puree; stir to combine. Refrigerate until cold, at least 2 hours (up to 4 hours).
Freeze peach mixture in an ice cream maker according to manufacturer's instructions. Transfer to a chilled airtight container; freeze at least 2 hours (up to 2 weeks). Let sorbet stand at room temperature 10 minutes before serving.