Line large baking sheet with aluminum foil.
Combine granulated sugar, brown sugar and rum in medium-size, heavy saucepan. Bring to boiling over medium-high heat. Continue cooking until temperature registers 275 degrees F on candy thermometer, 15 to 25 minutes (soft-crack stage; mixture forms pliable strands when drizzled from a metal spoon into bowl of cold water).
Stir in walnuts. Then quickly pour out onto prepared baking sheet, spreading out with metal spatula as you pour. Let cool completely.
Break into 2-inch pieces. Store in airtight container in cool, dry place for up to 2 months. Makes about 1-1/4 pounds.