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Heat vegetable oil in a medium-sized pot. Add onions, sauté about 2 minutes.
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Add leeks and garlic; sauté in pot for about 4-5 minutes until softened, but not browned.
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Add carrots; stir vegetable mixture until carrots begin to carmelize a bit, about 6 more minutes (watch to prevent over-cooking).
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Add potato, ginger, stock, and coconut milk (note- if using dairy cream, wait until soup is fully cooked, then add cream and heat to serving temperature); allow mixture to come to a boil for about 5 minutes; reduce heat and add spices (turmeric, paprika, salt, pepper).
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Cover and simmer on low until potatoes are tender, about 20-25 minutes. Puree soup in batches in the blender, or use a hand-held blending appliance to obtain a smooth, velvety textured soup.