1. The night before: place the dried lima beans in a large bowl and cover with several inches of cold water. Soak for at least 8 hours or overnight.
2. In the morning, drain and rinse the soaked beans. Transfer them to a large heavy saucepan (at least 4-quart / 3.8 L capacity) and add the 3 pints (1.4 L) of fresh cold water. Bring to a boil over medium-high heat, then reduce heat and simmer, partially covered, for 60–90 minutes, until the beans are very soft and crush easily between two fingers.
3. While the beans cook, cut the onion slices and carrot slices into small cubes (roughly 1/4 inch / 6 mm). In a medium skillet or saucepan, melt 2 tablespoons (28 g) of the butter over medium heat. Add the diced onion and carrot and cook for 5 minutes, stirring occasionally, until the vegetables are slightly softened and the edges of the onion turn translucent. Remove the vegetables with a slotted spoon and set aside.
4. Drain the cooked beans, reserving any remaining cooking liquid. Pass the beans through a fine-mesh sieve or food mill, pressing firmly with the back of a spoon, until you have a smooth puree. Return the puree to the large saucepan. Add enough reserved cooking liquid to bring the mixture to a pourable soup consistency. Bring the soup to a gentle boil over medium heat.
5. Make a beurre manie (a technique for thickening a liquid by whisking raw flour into butter): in a small bowl, work together the remaining 2 tablespoons (28 g) of butter and the 2 tablespoons (16 g) of flour with a fork until a smooth paste forms. Add the salt and pepper to the paste. Whisk this mixture into the boiling soup in small additions, stirring constantly, until fully incorporated and the soup has thickened slightly, about 2–3 minutes.
6. Stir in the cream or whole milk. Reheat the soup over medium-low heat for 3–5 minutes, until steaming and heated through — do not allow it to boil vigorously once the cream is added, as this may cause the soup to break.
7. Strain the finished soup through a fine-mesh sieve into a clean saucepan or serving vessel, pressing gently to extract all liquid. Discard solids. Add the remaining 2 tablespoons (28 g) of butter in small pieces, stirring until each piece is melted and incorporated. Taste and adjust seasoning. Serve immediately.
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