In a saucepan, heat the 2 tablespoons of oil; add the prosciutto and cook over low heat until the fat has rendered.

In a food processor, finely chop the carrot and celery. Add the vegetables to the saucepan along with the onion and garlic and cook, stirring, until the vegetables are softened, 7 minutes. Add the tomato paste and stir over moderately high heat until shiny, 1 minute. Add the lentils and 2 1/2 cups of the broth and bring to a boil. Simmer over low heat, stirring occasionally, until most of the broth has been absorbed, 25 minutes. Add 1 more cup of the broth and continue simmering until absorbed, 10 minutes. Add the remaining 1/2 cup of broth and simmer until the lentils are tender and suspended in a creamy sauce, 10 minutes longer. Season with salt and pepper.

In a large nonstick skillet, heat a thin film of olive oil. Crack the eggs into the skillet, season with salt and cook over moderately high heat until the edges are golden, the whites are just set and the yolks are still runny. Spoon the lentils into shallow bowls and top with the eggs and arugula. Grate the cheese over the eggs and drizzle with balsamic vinegar. Serve right away.