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Peel and dice the shallots.
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Peel the potatoes and cut into small pieces.
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Wash the radishes and pat them dry. Trim the greens and chop them into small pieces.
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Slice the red part of the radishes into rounds.
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Heat the unsalted butter in a large saucepan over medium-high heat.
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Add the shallots and stir fry until they become transparent.
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Add the chopped parsnip and potato pieces to the pan and stir to combine.
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Season with salt and freshly ground pepper.
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Cook the mixture, stirring occasionally for 3 minutes.
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Add the vegetable broth to the pan. Bring the mixture to a boil.
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Lower the heat and let the mixture simmer for 15 -20 minutes or until the vegetables have all become soft. You should be able to easily run a fork through them.
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Add the radish greens to the soup and cook for another 5 minutes.
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Stir the heavy cream into the soup and let cook another 5 minutes.
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Use a hand blender to puree the soup. If you do not have one handy, transfer the soup to a food processor or blender and puree, then transfer the pureed soup back into the saucepan.
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Add the radish rounds and the chopped fresh mint into the soup. If you prefer, reserve some of the red round pieces for garnish.
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Cook the soup for another 5 minutes.
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Remove from heat and divide into 4 bowls.
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Scatter the fresh red radish rounds and chopped mint or mint leaves over the top of the soup in each bowl and serve.