Step 1: Place onions, bay leaf, carrots, parsley root, tomatoes, cold water and 3 tablespoons butter in a 3-4 quart pot.
Step 2: Bring to a boil over high heat, stirring constantly.
Step 3: Reduce the heat to moderate, partially cover, and simmer for 30 minutes. Do not stir.
Step 4: Strain over a large bowl, pressing down on vegetables to extract all liquid.
Step 5: Discard vegetables. Return liquid to pot.
Step 6: Add fish and bring to a boil over high heat.
Step 7: Reduce heat to low, and simmer until the fish is opaque and firm to the touch, about 6-8 minutes.
Step 8: Remove fish and keep warm.
Step 9: Bring liquid in pot to a boil over high heat.
Step 10: Boil, uncovered, until reduced to about 1 1/2 cups.
Step 11: Place remaining butter in skillet and melt over high heat.
Step 12: When butter has almost stopped foaming, add mushrooms.
Step 13: Reduce the heat to moderate and cook, stirring occasionally, until mushrooms are soft and most of the juices have cooked away, 3-5 minutes.
Step 14: Stir in the reduced liquid and remove from heat.
Step 15: Whisk in heavy cream, one tablespoon at a time.
Step 16: Stir in the capers and olives. Serve over the fish.
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