Add potatoes to a pot of salted water and bring to a boil. Boil for 12 to 15 minutes until the potato slides out of a knife smoothly. Drain the water from the potatoes and add them back into the pot. 

Mash with a potato masher and while the potatoes are still warm, stir in vegan butter, gluten-free flour, chives, Korean chili powder, sea salt and ground black pepper. The mashed potato will start to come together almost like a dough. Set aside and let the mixture cool until you can handle it with your hands.

To make the filling, in a cast iron pan over medium heat, add vegetable oil, mushrooms, onions and sea salt. Cook for 4 to 5 minutes until the liquid evaporates and the mushrooms and onions start to turn golden brown. Add carrot and garlic and cook for 4 to 5 more minutes. Add this mixture to a medium bowl. Add chopped kimchi and chives. Set aside. Wipe the cast iron clean with a paper towel to fry the stuffed potato cakes later. 

To make the gochujang aioli, in a small bowl add warm kimchi brine or hot water, warm lemon juice and add gochujang paste. Whisk until smooth. Whisk in vegan mayonnaise, garlic, chives and black pepper. Set aside.

To assemble your potato cakes, use a 1/4 measuring cup and your hands to scoop up some mashed potato mixture. Release the potato mixture from the 1/4 cup so it holds its shape and then cut it in half horizontally, like you’re slicing a burger bun. This gives you the bottom and top halves of your potato cake. 

Use a tablespoon to measure out a heaping tablespoon of kimchi mushroom filling and place on one half of the potato mixture, flattening gently, so it doesn’t go all the way to the edge. The potato cake should be about 3-inches in diameter. 

Add a teaspoon of gochujang aioli in the center, slightly spreading with the back of the teaspoon but not all the way to the edges. Place the other half of the potato cake on top and use your fingers to push down the edges of the potato cake to seal. Repeat until all the potato mixture is used up. If any of the gochujang sauce is coming through, take a little potato mixture and seal up the hole!

Heat up the cast iron pan over medium heat. Add 1 teaspoon of vegetable oil. You will fry your potato cakes in batches. Add the potato cakes to the hot oil to pan fry and use a spatula to gently flatten the top slightly. Pan fry for 3 to 4 minutes on each side, until golden brown. Repeat until all the potato cakes are cooked. Transfer cooked potato cakes to a cooling rack lined with paper towel. 

For serving, you can stir in another 1 tablespoon of finely chopped kimchi into the remaining gochujang aioli for dipping or spoon the extra aioli on the potato cakes. 

Garnish with chives and serve immediately.