Heat olive oil in a large sauté pan over medium-high heat until hot. Add the onion and cook until soft but not brown, 2 to 3 minutes. Add the thyme; stir, and add 2 cups of the simmering broth. Bring to a boil.
Add the pastina; stir well, and reduce the heat to maintain a slow simmer. Season with salt and pepper. Add simmering broth 1/2 cup at a time as the previous addition is absorbed, and stirring occasionally to prevent the pastina from sticking to the bottom of the pan, until the pasta is al dente, about 15 minutes.
Add the squash and turkey to reheat. Stir well. The consistency should be quite loose, like a thick soup. Add more broth if necessary. Add the 1/2 cup cheese and let melt for a moment before stirring inches Taste for seasoning. If desired, pour into hollowed-out pumpkin for serving; garnish with cheese.