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Coat the bottom of a large saucepan or Dutch oven with 1 tablespoon of olive oil. Place over medium heat. Add white parts of leeks and garlic. Sautée until leeks are softened, about 5 minutes.
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Add wine and raise heat to medium-high. Bring to a simmer, Allow to simmer, stirring occasionally, until reduced by half, about 5 minutes.
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Add broth and cauliflower. Bring back to a simmer and allow to cook until cauliflower is tender, about 10 minutes.
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While cauliflower simmers, coat a large skillet with remaining 1 tablespoon of olive oil and place over medium heat. Add mushrooms in as even a layer as you can get them and cook for 5, minutes, until lightly browned on bottoms. Gently flip and cook another 5 minutes on opposite sides. Remove from heat and transfer to a plate.
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Working in batches if needed, transfer cauliflower mixture to blender or food processor and blend until completely smooth. This may take a few minutes. Return to pot.
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Add milk, salt and pepper to cauliflower mixture. Place over medium-high heat and cook just until heated throughout. Stir in mushrooms and green parts of leeks.