In a large pot, place potatoes and add enough water to cover. Cook 13 to 15 minutes or until very tender. Drain potatoes in colander. Return cooked potatoes to pot. With potato masher, mash in hot milk, 4 tablespoons of butter, salt, and pepper. Beat with wooden spoon until potatoes are smooth and creamy. Set potatoes aside and keep warm.

Meanwhile, heat olive oil and butter in a large skillet over medium-high heat. Add onions and let stand for about 5 minutes. Cook for 15 minutes more, stirring until golden brown. Toss well. Add water and balsamic vinegar, stirring well. Continue cooking until completely reduced and deep golden brown. Pour caramelized onions into the hot mashed potatoes and stir. Finally, fold in the goat cheese. Serve warm.