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For the beurre blanc: Put the cream, lemon juice, vinegar and shallots in a saucepan over medium heat. Bring to a boil and reduce by about half. Whip in the butter until smooth. Strain the sauce to remove the shallots. Add salt and pepper to taste and set aside.
For the grouper: Heat a medium nonstick pan over medium heat and add the olive oil. Sprinkle Cajun seasoning over one side of the grouper (the serving side). Cook until well seared and cooked through, about 3 minutes per side. Transfer to a plate.
Heat a nonstick pan over medium heat. Add the olive oil, haricots verts and minced garlic and cook briefly. Add the linguine and toss in 2 ounces (1/4 cup) of the beurre blanc sauce.