Clean and trim the mushrooms, then halve or quarter them lengthwise to accentuate the natural shapes of the different mushrooms.
Score the duck skin in a crosshatch pattern with a small sharp knife. Season with 1 1/2 teaspoons salt and 3/4 teaspoon pepper.
Heat the oil in a medium skillet over medium-high heat until hot but not smoking, then add the mushrooms, garlic, 3/4 teaspoon salt and 1/4 teaspoon pepper. Cook, stirring occasionally until the mushrooms are golden and any liquid they give off evaporates, 6 to 8 minutes. Add the broth, butter, lemon juice and parsley and cook, swirling the skillet, until the butter melts and the liquid forms a sauce. Keep warm.
Heat a large heavy skillet over medium-high heat until hot, then add the duck skin-side down. Cook until the fat begins to render from the skin, about 2 minutes, and then reduce the heat to medium. Pour any excess fat from the pan as needed while cooking the duck (use a large spatula to hold the duck in place). Continue to cook the duck, uncovered, without turning, until the skin is golden brown and crisp, about 10 minutes. Turn the duck over and continue to cook until an instant-read thermometer inserted in the thickest part of the breast registers 135 degrees F for medium rare, about 1 minute more. Transfer the duck to a cutting board and let rest 5 minutes.
Slice each duck breast into 5 to 6 pieces diagonally. Divide the Carrot-Cauliflower Puree among 4 plates, then top with the sliced duck breasts. Spoon the mushrooms and sauce around the plates and serve.
3/4 pound carrots, cut into small pieces
3/4 pound cauliflower florets, trimmed and cut into small pieces
Kosher salt and freshly ground black pepper
1/2 cup heavy cream
Put the carrots and cauliflower in a heavy medium saucepan with 1 teaspoon salt, 1/2 teaspoon pepper and 1 cup water. Bring to a boil over medium heat, cover and lower the heat to a simmer. Cook until the vegetables are very tender, about 30 minutes.
Transfer the vegetables to a food processor. Add the heavy cream and process until very smooth. Return the puree to the saucepan, reheat over medium heat and season with salt and pepper. (The puree can be made up to a day ahead and reheated.)