In a heavy 5-quart kettle/pot over medium heat, bring the sugars, half-and-half or cream, and corn syrup to a boil, stirring until the sugars dissolve and the mixture begins to boil. Reduce the heat to medium low. Cover and cook 2 to 3 minutes to dissolve the sugar crystals on the sides of the pan. Remove the lid. Cook slowly, stirring only as needed to prevent scorching, to the soft ball stage (234°F to 240°F, with 234°F recommended).
Remove from the heat. Add the butter, and walnuts, without stirring or shaking the pan. Cool to lukewarm (110°F), about an hour. Butter a pan.
Add the vanilla. Beat by hand until the candy begins to thicken and lose its gloss. Pour into the buttered pan. Cool and cut into squares. Store in an airtight container.