Preheat oven to 425°F. Line a baking pan.
Add tomatoes, onion, and garlic to the lined pan. Coat with olive oil, salt, and sugar. Toss to mix.
Roast for 20-25 minutes, then let cool.
Add roasted tomatoes, onion, garlic (including juices), chicken broth, gochujang, and half a slice of bread to a blender.
Blend on medium-high until warm to the touch and well-mixed. Adjust seasonings to taste. Add more bread if needed for thickness.
Thread rice cakes onto skewers. Melt butter in a pan and cook skewers until golden brown and slightly crispy.
Top rice cake skewers with shredded mozzarella and serve immediately with the tomato soup.