Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until potatoes are soft and you can easily insert a fork into them, about 20 minutes. Drain water and pour potatoes into a large mixing bowl.
Preheat the oven to 350 degrees F (175 degrees C).
Stir milk, cream cheese, and butter into the potatoes. Mix with an electric mixer on medium speed until creamy; do not overmix. Add 3/4 cup Cheddar cheese, 3/4 cup mozzarella cheese, and chives. Mix with a mixing spoon.
Pour mixture into a large baking dish or 4 individual ramekins and top with remaining Cheddar cheese and mozzarella cheese.
Bake in the preheated oven until cheese on top is melted, 10 to 15 minutes. Turn the oven’s broiler on high and brown the top of the cheese, 2 to 3 minutes, making sure to watch as it can overbrown easily.