Preheat the oven to 350°. In a 9-by-13-inch baking dish, combine the squash with the sage sprig, 2 tablespoons of the butter and 1/4 cup of water; season with kosher salt and pepper. Roast for about 25 minutes, until the squash is tender and most of the liquid has evaporated. Discard the sage.

Transfer the squash to a food processor and let cool slightly. Puree until smooth and season with kosher salt and pepper. Scrape the squash puree into a medium saucepan and keep warm over very low heat.

Meanwhile, in another medium saucepan, bring the stock to a simmer. Keep warm over very low heat.

In a large saucepan, heat the 1/4 cup of olive oil. Add the onion and a generous pinch of kosher salt and cook over moderate heat, stirring, until softened, about 3 minutes. Add the rice and cook, stirring, until well coated with oil, about 1 minute. Add the wine and cook, stirring, until absorbed. Add 1 cup of the warm stock and cook over moderate heat, stirring constantly, until nearly absorbed. Continue adding the stock 1/2 cup at a time, stirring constantly and allowing the stock to be nearly absorbed between additions. The risotto is done when the rice is al dente and suspended in a creamy sauce, 35 to 40 minutes total. Stir in the 2 tablespoons of Parmigiano and the remaining 1 tablespoon of butter and season with kosher salt and pepper.

Spoon the squash puree into shallow bowls, then spoon the risotto on top and garnish with the burrata. Top with a sprinkle of flaky sea salt and a drizzle of olive oil and serve. Pass additional cheese at the table.