Whisk together the eggs and Parmesan in a small bowl. Set aside.
Heat the olive oil in a medium skillet over medium-high heat. Add the prosciutto and cook until beginning to brown and get crispy, 4 minutes. Add the shallots and cook, stirring with a wooden spoon, until softened and fragrant, 2 minutes. Stir in the orzo and toast for 3 minutes. Add 3 1/4 cups water and the salt, reduce the heat to medium low and simmer for 8 minutes, stirring often to prevent sticking. Stir in the tomatoes and peas. Cook until the orzo is al dente, another 3 minutes.
Off the heat, add the egg mixture to the skillet while stirring constantly with a wooden spoon. Continue stirring until the sauce has thickened slightly and coats the pasta. Sprinkle with the basil and serve.