Cut kabocha into 1 1/2 inch chunks. Cover and microwave for 3 minutes on high, until soft. Once cooled, cut off the skin.
Mash with a fork. Start with 4 tbsp of potato starch and mix together until well combined. Depending on the moisture content of the kabocha, you may need to add up to 8 tbsp of potato starch. Adding more potato starch will yield a firmer chewy mochi, whereas adding in less will make for a softer mochi.
Divide the mixture into 4-5 amounts. Oil your hands and then place one portion of the kabocha mash in your hands. Flatten and stuff with cheese (or other filling of choice). You can also just roll them into a ball and flatten them into a small pancake.
Heat vegan butter over medium heat. Once melted, add in the mochi and pan fry each side until golden brown (about 1 1/2-2 minutes per side). If the edges look uncooked, cover and reduce heat to low. Cook for 1-2 minutes or until edges are cooked.