For this recipe, you will need an accurate candy thermometer. You'll also need to cut three-inch square pieces of waxed paper or cellophane in which to wrap the caramels.
Lightly brush bottom and sides of a 9-by-13-inch rimmed baking sheet with oil. Line with parchment, leaving a 2-inch overhang on long sides; lightly brush parchment with oil.
Bring cream, sugar, butter, and corn syrup to a boil in a large saucepan over high heat, stirring until sugar dissolves. Reduce heat to medium-high; cook, stirring occasionally, until caramel reaches 248 degrees on a candy thermometer, about 15 minutes.
Immediately remove caramel from heat, and stir in salt, vanilla, nuts, and thyme. Pour caramel onto baking sheet, and let stand, uncovered, at room temperature at least 8 hours and up to 1 day.
Lifting by parchment overhang, transfer caramel to a large cutting board. Cut into 3/4-by-1 1/4-inch pieces; wrap each piece in waxed paper or cellophane.