Line 9x5-inch loaf pan with foil; spray foil with cooking spray.
In 1-quart saucepan, heat whipping cream, vanilla, kosher salt and 2 tablespoons of the butter over medium heat until mixture just begins to simmer. Remove from heat; cover with lid. Set aside.
In 2-quart heavy saucepan, mix sugar and corn syrup; add candy thermometer. Cook over medium heat, stirring gently, until sugar is dissolved. If granules of sugar become stuck to sides of saucepan, use wet pastry brush to brush granules down towards syrup. Cook syrup, without stirring, until thermometer reads 310°F. If hot spots or darker areas occur while cooking, gently roll saucepan around to evenly disperse syrup.
Remove saucepan from heat. Slowly pour cream mixture into syrup. Syrup will bubble violently so be careful to add cream slowly to ensure syrup does not bubble over edge of saucepan. Stir until smooth.
Return saucepan to medium heat. Cook caramel until thermometer reads 260°F.
Remove saucepan from heat; remove thermometer. Stir in remaining 2 tablespoons butter until caramel is smooth.
Pour caramel into loaf pan. Cool 8 to 10 minutes. Sprinkle coarse sea salt evenly over top. Cool completely in pan on cooling rack.
When caramel is cool, remove from pan by lifting foil. Peel foil from caramel. Using sharp long knife, cut into 35 to 40 squares. Wrap each caramel individually with waxed paper or cellophane.