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For the vegetables: Preheat the oven to 350 degrees F. Coat the carrots with olive oil and sprinkle with salt and pepper. Roast, covered, on a baking sheet until tender. Let cool, and then dice. Peel the sunchokes, placing them in water so they do not discolor. Slice, drain and saute over medium heat in clarified butter until tender, and then chill.
For the grouper: Turn the oven temperature up to 400 degrees F. Heat a saute pan until hot, then add clarified butter and heat until hot. Sprinkle the fish with salt and pepper, then sear the pretty side of the fish until begins to brown, then transfer to the oven for 7 to 9 minutes. Do not flip or move fish. Remove the pan and add the whole butter, garlic and thyme.
For the beurre blanc: Meanwhile, combine the wine, shallots, orange juice and thyme in a saucepan and cook until reduced by 95 percent (almost 1 tablespoon liquid remaining). Remove from the heat and whisk in the chilled butter to combine. Strain through a fine mesh sieve.
To serve: Preheat a saute pan. Saute the mushrooms in some clarified butter. Add the other vegetables and warm, adding a small amount of stock if needed to prevent the vegetables drying out.
Baste the fish with the melted butter and pan drippings. Arrange with buerre blanc on the bottom, then vegetables, then grouper, and garnish with pea puree.