Preheat oven to 350 degrees Fahrenheit.

Prepare the chestnuts by making an incision through the shell of each chestnut.

Roast the chestnuts until tender, about 35 minutes.

Turn the oven off and leave the chestnuts in the oven.

Remove chestnuts a few at a time, peel while hot, and discard shells and inner skin.

Break the chestnuts into halves or quarters and set aside.

Place a large pot of water over high heat and bring to a boil.

In a large saute pan, melt butter and olive oil over medium heat.

Add the leeks and garlic, and cook until softened, about 6 minutes.

Add the roasted chestnuts and butternut squash, season with salt and pepper.

Cook until the squash is tender, about 15 to 20 minutes.

Add vermouth and chicken stock, raise heat to medium-high and cook until liquid is reduced by a little more than half, about 2 minutes.

Add salt and pasta to the boiling water and cook until al dente, 8 to 10 minutes.

Drain the pasta and add it to the saute pan along with half of the goat cheese, and toss to combine.

Transfer the pasta mixture to a serving dish.

Garnish with the remaining goat cheese and basil leaves.