Split the squash in half lengthwise and scrape out the seeds. Place the squash in a microwave-safe dish with about 1/2 inch of water; cover and microwave on high for about 15 minutes, until squash has softened all the way through. Let sit until cool enough to handle, and then use a fork to pull the insides of the squash into shreds. Transfer to a bowl, toss with 1 tablespoon of the oil, and season to taste with salt and pepper; keep warm.
Meanwhile, heat the olive oil with the butter in a large pan over medium heat. When the butter has melted, add the mushrooms and shallots. Cook, stirring, until the mushrooms have softened and released some of their liquid, about 5 minutes. Stir in the thyme, and season to taste with salt and pepper.
Pour in the cognac and chicken stock; stir to combine, then cook for about 3-4 minutes, until the liquid has partially evaporated. Remove from the heat, and stir in the cream and chives. Season to taste, adding a squeeze of lemon juice, if needed. Pour over the squash, toss to combine, and serve.