Baked in coarse salt to seal in the juices, these lemon-topped rounds of tender chicken reveal an asparagus-and-Parmesan stuffed center.
Prepare the chicken: Cut all breast meat from bone in one piece, leaving the skin intact. Cut off tenders from breast. Remove enough thigh meat (without skin) to measure about 1/2 pound; cut into 1-inch pieces, and refrigerate until ready to use. Using a sharp knife, butterfly the thickest part of the breast, and fold thin parts down so that breast is of even thickness all over. Cover with plastic wrap, and refrigerate until cold.
Meanwhile, make the filling: Bring a large stockpot of water to a boil; add 1 tablespoon salt. Prepare an ice-water bath; set aside. Add asparagus to the boiling water, and cook until asparagus is bright green and tender, 2 to 3 minutes. Plunge asparagus into ice-water bath to stop the cooking. Drain, and pat dry with paper towels; set aside.
Heat oil in a medium skillet over medium-low heat. Add leek and garlic; season with salt and pepper. Cook, stirring occasionally, until leek is soft and translucent, about 5 minutes. Remove from heat, and let stand until cool.
Transfer the leek mixture to a food processor; pulse until coarsely pureed. Add reserved thigh meat, and pulse to combine. Add the cream and 1 teaspoon salt; pulse. Add the Parmesan cheese, breadcrumbs, and lemon juice, and pulse until filling is fully combined and finely chopped (you should have about 1 cup filling); set filling aside.
Preheat oven to 400 degrees. Stuff the chicken: Using your fingers, loosen skin from breast meat. Slide 3 lemon slices and 1 tarragon sprig between skin and meat of each side of breast. Lay breast, skin side down, on a double layer of cheesecloth (about 18 by 24 inches). Spread 3/4 cup reserved filling over center of breast. Arrange about 10 asparagus spears over filling; reserve remaining asparagus for serving. Spread remaining 1/4 cup filling over top of asparagus.
Fold both sides of breast toward center to enclose filling; secure with toothpicks. Cover any skinless areas of meat with remaining lemon slices. Wrap chicken tightly in the cheesecloth, and tie up with kitchen twine, forming a log (about 9 inches long and 5 inches in diameter). The chicken can be prepared ahead up to this point and refrigerated, wrapped in plastic, up to 1 day.
Make the crust: Using your hands, combine salt, egg whites, and 2 tablespoons water. Stir in tarragon leaves. Using about 1/2 cup of the salt mixture, make a bed in a large baking dish. Place cheesecloth-wrapped chicken, seam side down, on top of salt. Pat remaining salt mixture over chicken, covering it completely with a 1/2-inch-thick layer.
Bake the chicken until an instant-read thermometer inserted through the crust into the center of filling registers 155 degrees, about 1 hour 10 minutes (begin checking after 1 hour). Remove chicken from oven; let stand at room temperature until thermometer reaches 165 degrees, about 10 minutes.
Break off crust; discard. Let chicken stand for at least 30 minutes before slicing. Remove cheesecloth, and discard. Transfer chicken to a cutting board, and cut into 1/2-inch-thick slices. Drizzle remaining asparagus with oil and lemon juice, and sprinkle with salt. Serve chicken warm or at room temperature with the asparagus. Serve wild mushrooms with red onions and scallions on the side.
Use the remaining chicken bones and meat to make stock. You can ask your butcher to remove the chicken breast from the bone, or do it yourself.