Divide the 4 pieces of mozzarella on 4 slices of bread. Cover with the 4 remaining slices of bread to form sandwiches.
In a wide, shallow bowl, add the eggs, milk, thyme, and salt and mix until well blended. Dip the sandwiches into the egg mixture one by one to coat, and set aside.
In a large nonstick sauté pan, add 1 tablespoon of olive oil and 1 tablespoon of butter. To the melted butter, 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.