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Over medium heat add the olive oil to a large pot. Add the caraway seeds and let them cook until a few start to pop.
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Add in the onion and fennel, and cook them for about 10 minutes, until they are lightly golden. Stir frequently.
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Add the beets, 1 teaspoon of salt, and vegetable stock. Bring the soup to a boil over high heat, then lower the heat to a simmer and cover the pot with a lid. Cook for about 25-30 minutes until the beets are very tender.
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Meanwhile, toast the pistachios in a dry pan over medium-low heat until fragrant, about 5 minutes.
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Process the nuts in a food processor with garlic and beet greens until a chunky paste forms. Slowly stream in the 1/4 cup of olive oil until blended. Stir in 1/2 teaspoon of salt. Set aside.
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Turn off the heat for the soup and process with an immersion blender or in a blender in batches (be careful, it is hot and can expand!) until smooth.
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Stir in the creme fraiche. Serve the soup in bowls and top with another dollop of creme fraiche and spoon full of pesto.