Fill a large pot with water, add a generous pinch of salt, and bring to a boil over high heat.
Add chopped potatoes and cook until fork tender.
Drain thoroughly and transfer to a bowl. Mash until smooth with no lumps.
Add potato starch and salt. Mix until fully combined and a soft dough forms.
Divide the mixture into 6–8 equal portions, depending on desired size.
Flatten one portion in your palm. Place a small piece of cheese in the center and wrap the potato mixture around it, forming a smooth ball. Gently flatten into a thick disc. Repeat with remaining portions.
Heat oil in a large pan over medium heat.
Cook the Potato Mochi on both sides until golden brown and crispy, about 3–4 minutes per side.
For the sauce, combine soy sauce, water, brown sugar, and butter in a separate pan. Heat over medium until slightly thickened.
Add the cooked mochi into the sauce and cook for 1–2 minutes on each side, allowing them to absorb the glaze.
Serve Potato Mochi warm while the cheese center is still melty.