In a medium skillet, cook the walnuts, brown sugar, salt, cayenne and 1 teaspoon of walnut oil over medium-high heat. Cook until the sugar caramelizes and coats the nuts, about 5 minutes, stirring occasionally to separate the nuts. Transfer to a baking sheet covered with waxed paper to cool. (Note: Caramelized nuts will keep in a tightly sealed container at room temperature for several days.)