Trim the tough bottoms off the asparagus and cut the rest of the spears into 1-inch pieces.
Melt the butter in a 3- to 4-quart saucepan over medium-low heat. Add the shallots, and cook, stirring occasionally, until soft but not colored, about 5 minutes. Add the cream, 2 cups of water, 2 tsp. salt, and 1/2 tsp. white pepper. Increase the heat to high and bring the liquid to a boil. Add the asparagus, lower to a simmer, and cook until the asparagus is tender (taste a piece to see) but still quite green, about 5 minutes.
Purée the soup in batches in a blender and pass it through a fine strainer, pressing on the solids, into a bowl (if you plan to serve it cold) or into a clean saucepan.
To serve, reheat the soup if serving it warm. Stir in the lemon juice. Season to taste with salt and pepper and more lemon juice, if you like. Serve the soup in cups or small bowls, topped with the croutons and chives.
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