Grease a 9 x 13-inch baking sheet pan well with vegetable oil. Line the pan with parchment paper and grease paper as well. Set aside.
In a 4-quart heavy-bottomed saucepan, bring heavy cream, sugar, butter, and corn syrup to a boil. Continue to boil until 248°F is reached on a candy or thermometer. Swirl the pan every so often to heat evenly.
When the temperature reaches 248°F, remove from heat and stir in salt and vanilla extract. Immediately pour into prepared pan.
½ tsp kosher salt, 1 tsp vanilla extract
Cool 6-8 hours or up to 24 hours to allow caramel to set up.
Use parchment sling to remove caramels from pan. Remove caramels from parchment and cut with a large sharp knife. The caramels are slightly sticky, so to keep the shape and keep from sticking to each other, individually wrap in parchment papers or wax paper.
For turtle caramels
Sprinkle the pecans over hot caramel. Let cool. Drizzle over the melted semi-sweet chocolate. Let the chocolate cool before cutting.
1 cup pecans, ½ cup semi-sweet chocolate
For salted caramels
Let the caramels cool for 30 minutes, then sprinkle the flaky sea salt over the caramels. Finish cooling and cut.