Grease a 9-inch square pan with 1 teaspoon butter.
Combine white chocolate chips, sweetened condensed milk, and 1/3 cup butter in a microwave-safe bowl. Heat in the microwave until white chocolate starts to melt, about 3 minutes. Stir until smooth. Continue to microwave in 30-second intervals, stirring after each interval, until chocolate is completely melted.
Stir rum flavoring, vanilla extract, and nutmeg into the white chocolate mixture. Fold in cranberries and walnut pieces. Pour fudge into the buttered baking pan, spreading it evenly into the corners.
