Divide the 4 pieces of mozzarella on 4 slices of bread. Cover with the 4 remaining slices of bread to form sandwiches. Cut the bread away to leave a 1/4-inch border around the perimeter of the cheese and set aside.
In a wide, shallow bowl, whisk the eggs and add the milk, thyme leaves and salt and mix until well blended. Dip the sandwiches into the egg mixture one by one to coat, and set aside.
In a 12 to 14-inch non-stick saute pan, heat 1 tablespoon of olive oil until smoking. Add 1 tablespoon of butter and cook until the sizzling subsides. Place 2 sandwiches in the pan and cook until golden brown on the first side. Flip each sandwich and cook on the other side. Remove sandwiches and set aside. Repeat the process with the remaining oil, butter and sandwiches. Serve immediately.
