Clean the outside of the potatoes under cool running water to remove any dirt or grit. Cut the potatoes into fourths. Drop the potatoes into a bowl filled with ice water.

Bring a large pot of salted water to a boil. Drain potatoes and add to the boiling water. Boil for 20 to 25 minutes until fork tender.

While the potatoes boil, add 1 tablespoon olive oil to a medium skillet. Heat oil over medium heat for about a minute or so. Halve leek and run under cold running water to remove grit. Slice the whites of leek into ¼ inch wide slices. Add the slices of the leek to the hot pan. Sauté leeks for 7 to 10 minutes until tender. Add the grated garlic, stir, and cook for another minute.

When the potatoes are fork tender, drain and then return to pot. Add the leeks, remaining olive oil, and buttermilk/whole milk. Mash potato mixture until it reaches the desired consistency. Add the horseradish, cheese, salt, and pepper—stir to combine. Add more olive oil or milk as needed. The longer you keep them warming, the more liquid they will absorb.

Serve potatoes right away topped with more grated cheese.