Melt butter in a cast-iron skillet over low heat. Add prosciutto and cook until crisp, 3 to 5 minutes.
Beat eggs and cream in a cup and pour into the skillet. Cook over low heat while stirring until eggs start to set and get creamy, about 10 minutes. Stir in Parmesan cheese and cook and stir for 2 to 3 minutes. Mix in pesto and season with salt and pepper. Serve immediately.