Line an 8-inch-square baking pan with parchment paper, allowing it to drape over 2 sides, then brush the paper lightly with oil.
In a deep saucepan (6 inches wide and 4 1/2 inches deep), combine 1/4 cup water, the sugar and corn syrup and bring them to a boil over medium-high heat. Boil until the mixture is a warm golden brown. Don't stir -- just swirl the pan.
In the meantime, in a small pot, bring the cream, butter and 1 teaspoon of kosher salt to a simmer over medium heat. Turn off the heat and set aside.
When the sugar mixture is done, turn off the heat and slowly add the cream mixture to the sugar mixture. Be careful -- it will bubble up violently. Stir in the vanilla with a wooden spoon and cook over medium-low heat for about 10 minutes, until the mixture reaches 248 degrees F (firm ball) on a candy thermometer.
Very carefully (it's hot!) pour the caramel into the prepared pan and refrigerate for a few hours, until firm.
When the caramel is cold, pry the sheet from the pan onto a cutting board. Cut the square in half.
Starting with a long side, roll the caramel up tightly into an 8-inch-long log.
Sprinkle the log with kosher salt, trim the ends and cut into 32 pieces. (Start by cutting the log in half, then continue cutting each piece in half until you have 32 equal pieces.) It's easier to cut the caramels if you brush the knife with flavorless oil like corn oil.
Cut glassine or parchment paper into 4-by-5-inch pieces and wrap each caramel individually, twisting the ends. Store in the refrigerator and serve the caramels chilled.