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 ¼ inch of the cheese.
In a wide, shallow bowl, whisk the 4 whole eggs and 3 yolks together. (Reserve the whites for a healthy omelet.) Add the milk, thyme, and salt and whisk well. Dip the sandwiches into the egg mixture 1 by 1 to coat.
In a large non-stick sauté pan over medium high heat, add 1 tablespoon of olive oil and wait until it’s smoking. Add 1 tablespoon of the butter and cook until the sizzling subsides. 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 the sandwiches and set aside in a warm oven. Repeat the process with the remaining oil, butter, and sandwiches. Serve immediately.