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 and vanilla. 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 ¾-by-1 ¼-inch pieces; wrap each piece in waxed paper or cellophane.
