Place potatoes into a large pot and cover with salted water; bring to a boil. Reduce heat to medium-low and simmer until tender, about 20 minutes. Drain.

Combine flour, buttermilk, water, and 1 teaspoon salt in a bowl. Whisk with a fork; when the dough thickens, start kneading by hand. Knead for at least 10 minutes. Cover dough and let rest at least 30 minutes.

Peel and mash the boiled potatoes. Add mozzarella cheese and mix well. Add salt to taste. Divide filling into 12 balls.

Divide dough into 12 balls. On a floured surface, stretch each dough piece into a small, flat circle. Place a filling ball in the center. Pull up edges of dough to encase filling and form a dough ball. Repeat for all.

Heat a heavy-bottomed frying pan over medium heat, without oil or grease. Melt butter separately. Roll out each dough ball into a thin circle about 1/4-inch thick. Be careful to prevent tears.

Place rolled hychin into the hot frying pan and cook until golden-brown spots appear, about 2 minutes on each side. Brush with butter after cooking.