Heat the oven to 400°F. Brush the cut sides of the squash with olive oil and season with 1/4 teaspoon salt and 1/8 teaspoon black pepper. Place the halves cut-side down on a baking sheet and roast until just starting to soften, about 25 minutes.
Meanwhile, combine the wild rice and water in a small saucepan. Bring to a boil, then cover and simmer until the rice is tender and the water is absorbed, following the package timing. Set aside.
In a large bowl, combine the ground turkey, chopped onion, breadcrumbs, egg, tomato paste, Worcestershire sauce, dried thyme, cooked wild rice, remaining 1/4 teaspoon salt, and remaining 1/8 teaspoon black pepper. Mix gently until evenly combined.
Turn the roasted squash halves cut-side up. Divide the turkey and rice mixture among them, mounding it into the cavities. Pour the chicken broth into the bottom of the baking sheet, around the squash.
Bake until the turkey filling is cooked through and the squash is tender when pierced with a knife, about 30 to 35 minutes. Let the squash rest for a few minutes before serving.