Place 2-4 slices of mozzarella on 4 pieces of bread. Top with other pieces of bread to form sandwiches.
In a wide shallow bowl, whisk the eggs. Add the half & half, thyme, salt and nutmeg. Whisk well to combine.
In a large nonstick pan, heat the 2 tablespoons of olive oil over medium-high heat. Add 1 tablespoon butter and cook until sizzling stops.
Dip 2 of the sandwiches into the egg mixture, turning to coat. Place sandwiches in the pan and cook until golden brown on the first side ( about 2 minutes). Flip over and brown on the other side.
Transfer sandwiches to a platter and repeat process with other 2 sandwiches.
Cut sandwiches in half, and serve immediately.