Top each of 4 slices of bread with a piece of mozzarella. Cover with the 4 remaining slices of bread to form sandwiches. Trim the bread to within 1/4-inch of the cheese and set aside.
In a wide, shallow bowl, whisk the eggs until smooth. Add the 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 12 to 14-inch nonstick saute pan, heat 1 tablespoon of the olive oil until smoking. Place 2 sandwiches in the pan and cook slowly until golden brown on the first side, about 2 minutes. Flip each sandwich and cook on the other side for 2 more minutes. Remove sandwiches and set aside in a warm oven. Repeat process with the remaining oil and sandwiches. Serve immediately.