In a large pot, combine the dried lima beans with 7 cups (1680 ml) of cold water and 1 teaspoon of salt.

Bring to a boil, then reduce the heat to low, cover, and simmer gently for 1 hour, skimming any foam that rises to the surface.

While the beans cook, prepare your vegetables. Trim the stalks and any blemished outer layers from the fennel, then wash thoroughly and chop into 1-inch chunks.

Peel and chop the onions into 1-inch chunks. Trim the green beans and cut them into 1-inch pieces. Scrape and thickly slice the carrots. Wash, trim, and halve or quarter the mushrooms depending on their size.

In a separate large stockpot or Dutch oven, melt the butter over medium heat.

Add the chopped fennel and onions and sauté for 10-12 minutes, stirring often, until they have softened and become fragrant.

Add the green beans, carrots, and mushrooms to the pot with the fennel and onions.

Stir for another 3-4 minutes. Gently pour in the partially cooked lima beans along with all their cooking liquid.

Add the remaining 2 cups (480 ml) of water and 1½ teaspoons of salt.

Stir to combine, bring to a simmer, then cover and cook over low heat for 45-50 minutes, or until all the vegetables are tender.

In a medium bowl, whisk the eggs until frothy. Slowly whisk in the fresh lemon juice.

To temper the sauce, carefully ladle about 1 cup (240 ml) of the hot broth from the stew into the egg-lemon mixture, whisking constantly to prevent the eggs from scrambling.

Pour this tempered sauce back into the stew pot.

Stir the stew gently over the lowest possible heat for 2-3 minutes until the broth thickens into a creamy sauce.

Do not let it boil.

Season with additional salt and pepper to taste, and serve immediately.