Chop the potatoes into 1-inch chunks. Add to a large pot and cover with water. Bring to a boil and cook for 15 minutes until soft.
Mash the potatoes using a potato masher or ricer. Stir in the butter until it melts. Add the sugar, salt, and glutinous rice flour. Knead until a smooth dough forms. The dough should not be sticky. If it's sticky, mix in a little glutinous rice flour at a time until it's no longer sticky.
With a rolling pin, roll out the dough until it's ½-inch thick. Use a cookie cutter to cut out circles. Re-roll the remaining dough and repeat this process until you run out of dough.
Before frying the mochi, I like to make the sauce. Combine the soy sauce, honey, and sugar in a small saucepan and cook on medium heat until it boils. Let the mixture boil for about 1 minute. The idea is to evaporate the moisture from the soy sauce so that it doesn't make the potato mochi soggy.
Heat the oil in a nonstick pan over medium heat. Add 3-4 mochis and cook each side for 3-4 minutes until golden brown and crispy. Transfer to a wire rack. Brush on the sauce and enjoy!