In a saucepan, over medium-low heat, stir the cream, milk, and 1/2 cup of the sugar until the sugar dissolves. Scrape the seeds from the vanilla bean into the cream, then add the empty bean (or add the extract). Cook, stirring occasionally, for 5 minutes. Remove from heat.
In a small bowl, combine the gelatin and 3 tablespoons of hot water. Let rest, stirring occasionally, until the gelatin dissolves completely, about 10 minutes. Stir 1/2 cup of the cream mixture into the gelatin. Pour it back into the remaining cream mixture, then add the salt and stir. Transfer to a large bowl and cover with plastic wrap, pressing the wrap against the surface of the cream; cool slightly. Refrigerate for 2 hours. Remove and discard the vanilla bean (if using) and stir to recombine. Divide the mixture among six 1/2-cup ramekins or cups placed on a baking sheet, filling each 2/3 full. Refrigerate until set, at least 4 and up to 24 hours.
Fill a medium bowl halfway with ice water. In a large pot, over low heat, dissolve the remaining 1 cup of sugar in 6 cups of water. Add the lemon zest and bring to a boil. Add the peaches, reduce heat to medium, and simmer for 10 minutes. Transfer the peaches to the ice water for 1 minute and slip off the skins; set aside. Add the herb sprig to the syrup and simmer until reduced to 2 cups, 35 to 40 minutes. Remove from heat and let cool slightly. Place the peaches in a bowl. Strain the syrup over the peaches and refrigerate for at least 2 and up to 8 hours. To serve, turn each panna cotta onto a plate, place a peach alongside it, and spoon the syrup over both.