Beat together the egg and milk in a shallow bowl. Add the thyme leaves and season with a pinch of salt and pepper.
Dip the slices of bread in the egg mixture, turning them over and allowing the mixture to soak in for a few moments. Sprinkle the Parmigiano-Reggiano cheese onto a plate and dip the soaked bread into the cheese to coat it lightly.
Heat the olive oil in a frying pan. Add the slices of bread and cook until golden, turing the slices over to cook both sides. At the same time, grill the cherry tomato bunches for a few minutes.
Lift the cooked bread onto four warm plates and arrange the Parma Ham and cherry tomatoes on top. Serve at once.