Place the rinsed fava beans in a large pot and add enough water to cover them by at least 2 inches.
Bring to a boil over high heat.
Reduce the heat to a steady simmer, cover the pot loosely, and cook for about 30 minutes, or until the beans begin to soften.
Add the chopped onion, carrot, celery, and a generous pinch of salt and pepper to the pot.
Stir to combine.
Continue to simmer, stirring occasionally, for another 30-45 minutes, or until the beans are very soft and falling apart.
Add more water if the mixture becomes too dry.
Drain the beans and vegetables, reserving a little of the cooking liquid.
Mash the beans to your desired consistency using a potato masher or a wooden spoon. For a smoother texture, use a food mill or ricer.
Add a splash of the reserved cooking liquid if needed to reach a creamy consistency.
Taste and adjust the seasoning with more salt and pepper. To serve, transfer the mashed favas to a bowl and drizzle generously with the extra virgin olive oil and fresh lemon juice.