3 tablespoons olive oil
1 delicata squash, halved lengthwise, seeds removed, sliced crosswise ½ inch thick
6 garlic cloves, smashed
4 sprigs thyme, plus leaves for serving
Kosher salt, freshly ground pepper
12 ounces linguine
3 ounces Parmesan, finely grated (about 3 cups), plus more for serving