Place squash in a bowl; add onion, 1/2 of the balsamic vinegar, brown sugar, vegetable oil, salt, and pepper and 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.
Combine kale and broth in a heavy-bottom skillet; bring to a light simmer. Cover skillet, reduce heat, and cook until kale is tender but still bright green, about 5 minutes; drain any excess liquid. Mix kale, squash mixture, and remaining balsamic vinegar together in a serving bowl.
