How to Make It
In a large saucepan, melt 3 tablespoons of the butter. Add the leeks and cook over moderate heat, stirring occasionally, until just tender, about 5 minutes. Add the wine and chopped tarragon and cook until the wine evaporates, 2 to 3 minutes. Add the cream and season with salt and pepper. Cover partially and cook over low heat until the cream is slightly reduced, about 10 minutes. Scrape the mixture into a blender and puree until smooth. Return the creamy leek sauce to the saucepan and season with salt and pepper. Cover and keep warm.
Melt 1 tablespoon of the butter in a small skillet. Add the shredded beet, season with salt and pepper and cook over moderately high heat, stirring constantly, until tender, about 8 minutes. Stir in the lemon juice.
In a medium saucepan, melt the remaining 4 tablespoons of butter in the water over moderately low heat, whisking constantly. Add the lobster meat and cook just until heated through, 1 to 2 minutes. Keep warm.
Meanwhile, cook the pasta in a large pot of boiling salted water until barely al dente. Drain, reserving 1 cup of the pasta cooking water. Return the pasta to the pot and add the leek sauce, reserved pasta water and the butter from the lobster; season with salt and pepper and toss to coat. Transfer the pasta to plates or shallow bowls and top with the lobster meat. Garnish with the beets and tarragon leaves and serve at once.