Chop green onions, reserving about 1 tablespoon chopped green onion tops for garnish. Heat vegetable oil in a Dutch oven or heavy saucepan over medium heat; saute larger portion of green onions, tomato paste, red curry powder, cumin, and red curry paste in the hot oil until green onions and seasonings sizzle and give off a toasted smell, about 5 minutes. Stir garlic into green onion mixture; saute until garlic is fragrant, about 1 more minute.
Pour coconut milk into Dutch oven and stir until thoroughly combined. Mix fish sauce and brown sugar into coconut milk sauce, stirring to dissolve brown sugar. Bring sauce to a simmer.
Stir butternut squash cubes into sauce, turn heat to medium-low, and cover with a lid. Gently stir every 5 minutes until squash are tender, about 20 minutes. Watch carefully, squash overcooks easily. Stir basil leaves into butternut squash. Taste for salt and spiciness and adjust seasoning if needed. Transfer to a serving dish and sprinkle with reserved green onions.