Add 1 tablespoon olive oil to the pot and raise the heat to medium-high. Add garlic, ginger and curry powder and cook, stirring constantly, until fragrant, about 30 seconds. Add half of the kale and stir until it's beginning to wilt, about 1 minute. Stir in remaining greens, broth, all but ½ cup coconut milk and ½ teaspoon salt.
Meanwhile, warm 2 tablespoons olive oil in a Dutch oven over medium heat until shimmering. Add onion and cook, stirring frequently, until softened, about 5 minutes. Add sweet potato or squash, cover and cook, stirring occasionally, until the sweet potato is bright orange (or until the butternut is just beginning to brown), about 5 minutes. Transfer the mixture to a bowl for now.
To cook the rice, bring a large pot of water to boil. Add the rinsed rice and cook for 30 minutes, reducing heat as necessary to prevent overflow but keep the water boiling. Turn off the heat, drain the rice and return the rice to pot. Cover and let the rice rest for 10 minutes or longer, until you're ready to serve.