Heat the oil in a large, high-sided skillet until simmering. Add the onion, carrot, celery, garlic, 1/2 teaspoon of the salt and pepper. Cook, stirring occasionally, until the vegetables are soft, about 5 minutes. Add the sausage, season with the remaining 1/2 teaspoon salt, and cook, breaking up the meat with a wooden spoon, until browned and no longer pink in the middle, 6 to 8 minutes.
Pour in the wine to deglaze the pan, scraping up any browned bits from the bottom. Cook until the wine is almost fully evaporated.
Add the tomato paste and stir to coat. Add the tomatoes and their juices, broth, bay leaves, thyme, rosemary, and pepper flakes. Bring to a boil, then reduce the heat to low and simmer, stirring occasionally, until the flavors meld, about 30 minutes. Meanwhile, make the polenta
Bring the water to a boil in a medium saucepan over medium-high heat. Whisk in the salt and cornmeal. Reduce the heat to low and whisk the polenta continuously until slightly thickened and no longer sticking to the bottom of the pan, 2 to 3 minutes.
Cover and continue cooking, stirring and scraping the sides of the pan every 10 minutes, until the polenta is soft and creamy, 25 to 30 minutes total. Remove from the heat and stir in the cheese and butter until fully melted.
Remove and discard the bay leaves and thyme and rosemary sprigs from the ragù. Serve the ragù over the polenta.