Measure 1/4 cup plus 2 tablespoons heavy cream into a heatproof 1-cup measuring cup or small bowl. Stir in 1/8 teaspoon kosher salt until dissolved. Set aside.
In a small saucepan combine 1/4 cup water and 2 tablespoons light corn syrup. Pour 1 1/4 cups white sugar into the center of the pan, taking care that none of the sugar touches the sides.
Place the saucepan over medium-high heat. Gently stir only until the sugar dissolves and the mixture at the edges begins to boil, then stop stirring.
Cover the pan with a tight-fitting lid and let it boil 1–2 minutes, until the syrup looks completely clear (this helps dissolve any crystals on the sides).
Remove the lid and continue to cook uncovered, without stirring. Occasionally swirl the pan to promote even browning, and cook until the syrup is a light-to-medium amber color and reads about 320–340°F on a candy thermometer (if you don’t have a thermometer, watch for a deep golden-amber color).
Remove the pan from the heat. Carefully and slowly pour the reserved cream-and-salt mixture into the hot caramel while stirring constantly—the mixture will bubble vigorously. Keep stirring and scrape the bottom and sides until the bubbling subsides.
After bubbling has calmed, stir in 2 tablespoons butter and 1 teaspoon vanilla extract until the caramel is smooth. Let cool slightly, then set aside and proceed to make the brownies.