Combine the sugar and water in a heavy-bottomed sauce pan. Briefly stir to combine.
Turn the burner on to medium-high heat. Then — this is very important — do not stir or touch the pan until the sugar has caramelized. The sugar mixture will go from being completely clear, to a champagne color, to a light golden color, and then to increasingly darker shades of amber. (Keep a close eye on the sugar, as it darkens quickly once it begins to caramelize.) Once the sugar reaches a deep amber color (about the color of a copper penny), remove the saucepan from the heat and turn the burner off.
Right away, begin to slowly and carefully pour the heavy cream into the sugar mixture, whisking as you pour to quickly incorporate the cream. (The mixture will bubble furiously when the cream hits the hot sugar, so be very careful!) Add in the butter and whisk until combined. Add in the vanilla, bourbon and sea sea salt, and whisk until combined.
Meanwhile halve and pit the peaches, brush with butter and place pit-side down onto your grill or grill pan with tongs. Grill for about a minute per peach, or until nice grill marks appear.
Drizzle the sauce on your grilled peaches, serve over ice cream, gelato, or a thick Greek yogurt. To store the sauce, let it cool until it reaches room temperature (it will thicken considerably as it cools), then transfer the sauce to a sealed container and refrigerate for up to 2 weeks. Leftover grilled peaches are great tossed in a smoothie!