Add the olive oil to a large skillet and heat over medium-high. Once hot, add the sausage. Continue to cook the sausage until it is cooked through, breaking it down into small pieces as it cooks.
Use a slotted spoon to remove the sausage from the pan.
Add the butter, and lower the heat to medium. Once the butter has melted, add the onion, celery, garlic, salt, pepper, thyme, and sage. Continue to cook until the onion is translucent and the veggies are very tender.
Stir in the tomato paste and give it a minute to cook with the rest of the veggies.
Add the sweet vermouth, turn the heat up to medium-high, and let it cook for 3 minutes to let the alcohol burn off.
Add the chicken stock, butternut squash, and kale. Bring to a low simmer, cover with a fitted lid, and let cook until the squash is tender, about 8 minutes.
Remove the lid, add back the cooked sausage, and lower the heat to low. The goal is to keep the veggies warm while you make the polenta.
Add 6 cups of water to a medium-sized pot. Bring to a boil and add 1 teaspoon of kosher salt.
Once the water is boiling, slowly whisk in the polenta, then immediately lower the water to a low simmer. Cook for 5 minutes uncovered, whisking occasionally.
Remove from heat and whisk in the heavy cream and butter.
Pour the polenta onto the serving dish and top with the sausage-veggie mixture.
Serve warm!