In large skillet set over low heat, melt butter. Cook onion, stirring often, for 8 to 10 minutes or until softened and starting to brown. Stir in Roasted Chicken Base and 1 tbsp water. Cook for 10 to 15 minutes or until onion is very tender and caramelized. Let cool completely.
Meanwhile, transfer eggs to bottom of large saucepan in single layer; pour in enough cold water to cover by at least 1 inch. Bring to a boil over high heat; remove from heat. Cover and let stand for 12 minutes; drain and rinse under cold water. Peel eggs and cut in half lengthwise.
Carefully transfer egg yolks to food processor; set egg white halves aside. Add cooled onion mixture, mayonnaise and Dijon to food processor; pulse until blended.
Spoon or pipe mixture into egg white halves. Sprinkle with chives.
