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Separate broccoli stalks from crowns. Peel stalks and thinly slice. Cut crowns into 1" florets. You should have about 8 cups chopped broccoli total.
In a heavy large pot over medium heat, melt butter, then swirl in oil. Add shallot and cook, stirring, until softened, 3 to 4 minutes. Add broccoli, potato, 1 teaspoon salt, and a pinch of black pepper and cook, stirring occasionally, until broccoli is bright green, about 5 minutes. Add broth and 2 cups water. Cover and simmer over medium-low heat until broccoli and potato are very soft, 10 to 15 minutes. Let cool 5 minutes.
Using an immersion blender or in a standard blender, puree soup with another 1 cup water (work in batches if necessary). Return to pot, stir in cream, then warm over low heat, loosening with more water if necessary.
Stir in lemon juice; season with salt and black pepper. Drizzle with more cream. Serve with lemon wedges alongside.