In a medium nonstick skillet, melt 1 tablespoon of butter over medium-high heat. Add the onion and sugar; saute 10 minutes or until caramelized. Remove from heat; stir in the vinegar, if desired.
Place the potatoes in a saucepan, and cover them with water. Bring to a boil. Reduce the heat; simmer 15 minutes or until tender.
Drain and return the potatoes to the pan. Add 3 tablespoons butter and milk; mash to your desired consistency. Cook for 2 minutes or until thoroughly heated, stirring constantly.
In a small bowl, combine the Dijon mustard with the remaining ingredients, stirring with a whisk until blended. Add the Dijon mustard mixture and caramelized onion mixture, stirring to combine. Then stir it all into the mashed potatoes.
Notes
If preparing this as a GLUTEN FREE recipe, just make sure that you are using a brand of mayonnaise that is designated as GF.