Lightly oil the inside of a 5-quart saucepan. Combine sugar and corn syrup in pan; cook over low heat, stirring constantly, until sugar is dissolved and mixture comes to a boil. Clip on candy thermometer; cook stirring occasionally, until thermometer reaches soft ball stage.
Add warm cream very slowly so that mixture never stops boiling. Cook again until thermometer reaches soft ball stage.
Add butter, bit by bit, so mixture never stops boiling. Stir a little to blend; let mixture cook to soft ball stage again. Remove from heat; add vanilla. Pour in a steady stream into lightly oiled 9x9x2-inch pan. Do not scrape pan.
Mark in 1-inch squares, but do not cut all the way through. After caramel has cooled completely, it can be turned out onto cutting board and cut into marked squares. Wrap individual pieces in plastic wrap or waxed paper; store in airtight container. Makes 81 1-inch pieces.