Popularized in colonial Williamsburg, this simple, silky soup gets brightened up with lemon juice and served with diced green apple for a touch of tartness and red pepper flakes for heat.
Heat oil in a large pot over medium. Add carrot and onion; season with salt. Cook, stirring occasionally, until vegetables are soft but have not taken on any color, about 5 minutes. Add peanuts, broth, and 3 cups water; bring to a boil. Reduce heat and simmer, uncovered, until peanuts are tender, about 1 hour, 20 minutes.
Working in batches (do not fill jar more than halfway), puree soup in a blender until smooth, about 1 minute. (Or use an immersion blender.) Strain through a fine-mesh sieve, pressing on solids to remove as much liquid as possible. Return to pot and reheat over low, if necessary.
Stir lemon juice into soup. Season with salt and pepper and serve, topped with apple, red-pepper flakes, and a drizzle of oil.