Melt 3 tablespoons butter in a large skillet over medium-high heat. Add the onions and cook for 5 minutes. Reduce heat to medium, and continue cooking, stirring frequently, until the onions have softened, and achieved a deep brown color, 15 to 25 minutes. Stir in vinegar and thyme, and cook for 5 minutes more.

While the onions are cooking, place cubed potatoes in large pot and cover with salted water. Bring to a boil, then reduce heat to medium-low, cover, and simmer until the potatoes can be pierced easily with a fork, about 20 minutes. Drain the potatoes in a colander, and allow to steam for a minute.

To finish the dish, transfer the potatoes to a serving bowl, and use a potato masher to mix with Dijon mustard, horseradish, salt and cracked black pepper until smooth. Mix in softened butter, caramelized onions, and half-and-half until onions are distributed evenly.