For the rosemary sugar and roasted peaches:
For the salted caramel sauce:
To make rosemary sugar: In a food processor, combine raw sugar and chopped rosemary (make sure the rosemary is completely dry if you've just rinsed it). Process for about 30 seconds, or until well integrated. (You won't need the full amount of rosemary sugar for one recipe, but it keeps well tightly covered, and it can be used in countless ways: to sweeten iced tea, to sprinkle over fresh fruit, etc.)
To pan-roast peaches: Preheat oven to 425° F. Liberally sprinkle cut sides of halved peaches with the rosemary sugar, and follow with a pinch of salt.
Meanwhile, melt butter in a cast iron skillet over medium-high heat. Arrange the peaches in the skillet, cut side-down. Cook for 3 to 5 minutes without disturbing, until the cut sides begin to brown. (Note: The peaches will throw off juice while they’re browning, but it will eventually thicken in the oven.)
Transfer the skillet to the oven and roast for about 10 minutes. Flip the peaches, drizzle with butter, and continue roasting another 5 to 10 minutes, or until tender. (The total cooking time will depend on the size and ripeness of your peaches.)
To make salted caramel sauce: Combine sugar, corn syrup (or golden syrup), and 1/4 cup water in a deep saucepan. Cook over medium-high heat, stirring once or twice, until the sugar has dissolved. Continue cooking without stirring (stirring can cause the sugar to crystallize), swirling the pan occasionally while the sugar cooks. If you notice any sugar on the sides of your sauce pan, brush it down with a wet pastry brush to dissolve the sugar crystals.
Cook until the sugar has reached a deep amber color, about 7 to 10 minutes total. Then, immediately take the saucepan off the burner. (Keep a close eye: Your caramel can go from perfect to burnt in a matter of seconds. If using a candy thermometer, cook the sugar until it reaches 350° F.)
Once the pan is off the heat, add the heavy cream (stand back because it will sputter upon hitting the caramel), and then whisk in the butter until smooth. If the caramel develops any lumps, place it back over low heat and whisk until smooth. Let stand for 3 minutes, then add the vanilla extract. Season the finished caramel with fleur de sel to taste. Tightly covered and refrigerated, the caramel sauce should keep for several weeks in the fridge; gently rewarm it before serving.
Note: When making caramel, I prefer the taste of golden syrup to that of corn syrup. If using golden syrup, I highly recommend the use of a candy thermometer. The golden syrup makes the sugar solution an amber color from the get-go, making it much, much harder to judge the stopping point by eye.