Using a sharp chef’s knife or cleaver, remove stem from squash. Peel tough outer skin and cut squash in half. Remove seeds and cut into large 2-inch-long pieces.
Heat oil in a large stockpot over medium heat. Add onion and cook, stirring occasionally, until it begins to brown. Add garlic and season with salt and pepper; cook about a minute more and then add sausage. Break sausage into small pieces and cook until browned, about 6 minutes.
Add squash, season again with salt and pepper, chili and cook until squash becomes soft around the edges, about 6 minutes. Add wine and scrape up any browned bits that have stuck to the bottom of the pot. Add broth and thyme and bring to a boil. Reduce heat and simmer until squash is soft, about 45 minutes.
Using the back of a spoon, press squash against the side of the pot to break it up into pasta-like strands. Remove thyme stems; season with more salt and pepper to taste. Stir in spinach leaves and serve.