Cook leeks: Melt 2 tbsp. butter in a large frying pan over medium heat. Add leeks, 1/2 tsp. salt, and the pepper. Cook until leeks are very soft and separated into rings, stirring occasionally, 12 to 15 minutes. Add water if needed to keep from scorching, 2 tbsp. at a time. Set aside.
Put potatoes in a large pot and add enough cold water to barely cover. Stir in 1 tsp. salt. Bring to a boil over high heat, covered. Reduce heat to medium and boil gently until potatoes are tender when pierced, about 15 minutes. Drain; return to pot, off the heat.
Mash hot potatoes with a potato masher until all large chunks are broken and potatoes are fluffy. Add milk, 2 tbsp. butter, and 1 1/2 tsp. salt and mash to blend. If leeks are cool, gently warm in microwave. Stir all but 1/2 cup of leeks into potatoes.
Transfer to a serving bowl and scatter remaining leeks over the top.
Make ahead: Leeks, up to 5 days, chilled. Mashed potatoes, up to 1 day, chilled; reheat in a microwave until hot (if they've stiffened up, add enough milk to make them creamy).