Preheat the oven to 400 degrees F (200 degrees C).
Place squash into a large bowl. Add onion, 1 ½ tablespoons vinegar, brown sugar, oil, salt, and pepper; toss to coat. Spread mixture in a single layer onto a baking sheet.
Bake in the preheated oven until squash is tender and golden, 30 to 40 minutes. Set aside.
Combine kale and broth in a heavy-bottom skillet over medium heat; bring to a light simmer. Cover the skillet, reduce heat, and cook until kale is tender but still bright green, about 5 minutes; drain any excess liquid.
Mix together kale, squash mixture, and remaining 1 ½ tablespoons vinegar in a serving bowl.