Preheat oven to 375°F. Cut Kabocha in half. Remove seeds. Cut into 1-inch vertical slices. See pic for reference.
Drizzle a bit of olive oil on a baking sheet. Place Kabocha slices, drizzle 2 tablespoon olive oil over it, sprinkle salt and pepper (aprox. 1/4 teaspoon of each). Toss it gently with hands to coat. Cover with foil. Bake at 375°F for 10 minutes. Remove foil; bake 15 more minutes or until pumpkin is tender and browned, turning once.
While the Kabocha roasts, thoroughly rinse the kale in cold water, soaking if necessary to remove grit. Tear the kale into chunks. Give it a good spin or shake to remove excess water and place it in a large salad bowl. Reserve.
Remove Kabocha from the oven. Let it cool off slightly. Peel the skin and cut into small cubes. Reserve.
Heat the remaining olive oil (2 tbsp) in a small skillet over medium-high heat. Add the onion and cook until translucent. Season with some salt (1/2 tsp) and pepper. Remove from heat, add the balsamic vinegar, and poor over the kale in the salad bowl. Toss, and toss, and toss some more until kale is a fully coated and a bit wilted. Add the Kabocha cubes and pomegranate tossing it slightly.
Finish by topping it with the Pecorino or Parmesan cheese. Serve it warm.