Butter a 9-inch square baking dish; set aside. Butter the sides of a heavy 2-quart saucepan.
Combine both sugars, cream, milk, and butter in saucepan. Cook over medium heat, stirring constantly, until sugars dissolve and mixture comes to a boil. Cook to soft-ball stage (238°F on candy thermometer), stirring only if necessary. Immediately remove from heat and cool to lukewarm (110°F). DO NOT STIR.
Add vanilla extract and beat vigorously until mixture is very thick and starts to lose its gloss. Quickly stir in nuts and spread into prepared baking dish. Score while warm, and cut when firm.