Root Vegetable Pot Pie with Puff Pastry

Name: Root Vegetable Pot Pie with Puff Pastry
Ingredients: 1/2 cup unsalted butter (divided use); 2 medium shallots (peeled and cut in half); 2 teaspoons granulated sugar; 2 large carrots (peeled and cut into 1/2-inch pieces); 1 turnip (peeled and cut into 1/2- inch pieces); 1 celery root (peeled and cut into 1/2-inch pieces); Olive oil cooking spray; 1/3 cup all-purpose flour; 2 1/2 cups milk (whole, 2 percent or skim); 2/3 to 1 cup (3 to 4 ounces) grated Parmesan cheese; 2 teaspoons fresh thyme leaves; 1/3 cup chopped fresh Italian parsley; salt (to taste); freshly ground black pepper (to taste); 1 large egg (lightly beaten); 1 Sheet puff pastry (frozen, thawed)