Step 1Combine carrot, shallots, mushrooms and wine in saucepan over low heat until steaming hot but not bubbling, 8 to 10 minutes. Reduce for 1 hour, then strain through a fine mesh sieve into a clean pan. Return to heat and reduce until wine measures about 1/2 cup, 1 1/2 to 2 hours. Be careful in later stages to keep heat turned very low to avoid scorching. Strain through a fine mesh sieve into a clean pan and set aside or refrigerate until ready to use.
Step 2Remove halibut from refrigerator 45 minutes before cooking. Season halibut liberally with salt a few minutes before cooking. In a hot pan over high heat, saute halibut in oil until one side is golden brown. Lower heat to medium, turn halibut and continue cooking 1 more minute. Transfer to a baking sheet tray and bake at 225 degrees just until fish begins to flake, 5 to 10 minutes.
Step 3Reheat sauce reduction over medium heat and add butter, 1 chunk at a time, whisking continuously until all butter is emulsified into reduction. Taste sauce and season with salt and pepper. Divide sauce among 6 plates, top with halibut and serve.