Arrange a rack in the middle of the oven and heat the oven to 425ºF.
Prepare the following, adding each to the same large bowl as it is completed: Thinly slice 5 medium scallions (about 1 cup); reserve 2 tablespoons of the dark green parts for garnish. Mince 3 garlic cloves. Grate 1 pound low-moisture mozzarella cheese on the large holes of a box grater (about 4 cups).
Finely chop 1 cup drained Napa cabbage kimchi. Melt 1 tablespoon unsalted butter in a medium skillet over medium heat. Add the kimchi and 1 teaspoon toasted sesame oil. Cook, stirring occasionally, until the kimchi is lightly browned and most of the moisture is evaporated, 2 to 3 minutes. Transfer to the bowl with the mozzarella.
Add ⅓ cup mayonnaise, ⅓ cup sour cream, 1 tablespoon gochugaru, 2 teaspoons soy sauce or tamari, and 1 teaspoon granulated sugar. Stir to combine. Transfer the mixture to an 8x8-inch broiler-safe baking dish or 10-inch cast iron skillet and spread into an even layer.
Bake until lightly browned on top and bubbling around the edges, 17 to 19 minutes. Switch the oven to broil and broil until the top is dark golden-brown in spots, 2 to 3 minutes. Let cool for 5 to 10 minutes before serving. Sprinkle with the reserved scallions and serve with chips, crackers, or crudités.
