Heat the olive oil in a large Dutch oven over medium-high heat. Add the ground turkey and cook, breaking it into crumbles, until no pink remains, about 5–6 minutes. Drain off any excess liquid.

Stir in the onion and butternut squash; cook until the onion softens, about 3 minutes. Add the garlic, chili powder, cumin, smoked paprika, oregano, salt, and pepper; cook 1 minute more, until fragrant.

Pour in the fire-roasted tomatoes with their juices, the black beans, and the chicken broth, scraping up any browned bits from the bottom of the pot. Bring to a lively simmer.

Reduce the heat to medium-low, partially cover, and simmer until the squash is fork-tender, about 8 minutes. Cut any oversized squash cubes in half before cooking so everything softens at the same pace.

Mash a few spoonfuls of squash directly in the pot with a potato masher or the back of a spoon — this is what thickens the chili without flour or cornstarch. Stir in the brown sugar if using, and adjust salt and pepper.

Ladle into bowls and finish with shredded cheddar, a dollop of sour cream, cilantro, and a squeeze of lime.