Whisk the mayonnaise, lemon juice, and zest in a small bowl to blend. Set the mayonnaise mixture aside.
Toast the bread on a panini grill or griddle until golden brown. Spread the mayonnaise mixture over each slice of toast. Arrange the prosciutto over the toasts, dividing equally. Top with the arugula, then the tomatoes. Drizzle the oil over the sandwiches, then sprinkle with salt and pepper. Cut the sandwiches into 3 equal pieces and serve.