Make BBQ sauce: In a large bowl, stir together all ingredients until smooth.
Strip leafy kale from stalky stems. Tear leaves into 2” pieces and finely chop stems into ⅓” pieces. Keep separate.
In a large pot over medium heat, heat 2 tablespoons oil. Add gochugaru and garlic and stir until fragrant, 30 seconds. Increase heat to medium-high, then add onion, tomato, pepper, kale stems, and salt. Cook, stirring occasionally, until tomatoes begin to caramelize, 10 to 12 minutes.
Add remaining 1 tablespoon oil, then add kale leaves in two batches, allowing the first batch to wilt before adding the second. Once kale is fully wilted, about 6 minutes, add broth and use a wooden spoon to scrape and deglaze the pan. Bring to a simmer, then add 1 cup barbecue sauce, stirring to combine evenly.
Cover pot with lid, reduce heat to low, and gently simmer until kale is completely tender, about 25 minutes.
Top with fried onions before serving.