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In a large pot or dutch oven, heat enough oil to cover the base over a medium-high heat. Add sliced leeks and a generous pinch of salt and cook, stirring occasionally for 6-8 minutes until the leeks are softened and fragrant. Add garlic and cook, stirring, for 1 minute.
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Add broth and white beans, bring to a boil, reduce to a simmer and cook for 10 minutes uncovered. Add broccoli, cover, and cook for 5 minutes. Taste for seasoning and adjust. Stir in spinach and parsley until wilted (about 1 minute).
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Remove from heat and blend with an immersion blender (or in a regular stand blender in batches) until smooth-ish. Taste for seasoning again. Adding a spritz of lemon and serve warm.