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, applying all of your strength so you will have to put less effort into rolling. Cover dough and let rest at least 30 minutes.
Peel and mash the boiled potatoes in a bowl. Add mozzarella cheese and combine well. Add salt to taste if the mixture is too bland. Divide filling into 12 balls; place on a plate or baking sheet and cover.
Divide dough into 12 balls; these will be smaller than the filling balls. Place on a plate or baking sheet and cover.
Stretch 1 piece of dough into a small, flat circle. Place a filling ball in the center; pull up the edges of the dough and secure to have a ball encased in dough. Repeat with remaining dough and filling.
Heat a heavy-bottomed frying pan over medium heat; use no oil or grease. Place melted butter in a bowl nearby.Sprinkle a work surface with a generous amount of flour. Carefully and gently start to roll out a ball; the dough will be hard to work at first. Be careful not to press too hard with the rolling pin to avoid rips; roll into a circle about 1/4-inch thick.
Place hychin into the hot pan. Fry until golden-brown spots appear, about 2 minutes. Pierce with a knife to deflate, if necessary. Turn over and cook on the other side, about 2 minutes more. Brush generously with butter and transfer to a plate. Repeat with remaining balls and stack cooked hychin on the plate.
Cut the whole hychin stack into 8 slices like a pie.
Traditionally the fresh brined cheese is homemade. You can substitute with other brined cheeses (halloumi, cotija). For best results pass the cheese through a meat grinder, but finely shredded works well too.
These go great with a simple sauce: combine plain yogurt, crushed garlic, salt, and chopped fresh dill.
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