Carrot and Spring Onion Toad in the Hole

Name: Carrot and Spring Onion Toad in the Hole
Ingredients: 2 tablespoons extra-virgin olive oil; 2 tablespoons unsalted butter; 6 ounces small multicolored carrots (about 10 carrots), trimmed; 6 ounces small spring onions (about 4 onions), green parts cut into 3-inch pieces and bulbs halved lengthwise; 4 (1-oz.) breakfast sausage links; 1 teaspoon finely chopped fresh thyme; 2 teaspoons kosher salt, divided; 1 teaspoon black pepper, divided; 3 large eggs; ¾ cup whole milk; Pinch of freshly grated nutmeg; ¾ cup all-purpose flour (about 3 1/4 oz.); Finely chopped fresh flat-leaf parsley, for garnish