For the Tangy Dill Sauce: in a medium bowl, whisk together crème fraîche or sour cream, yogurt, dill, chives, Dijon, lemon zest, and juice until well combined. Season to taste with salt and pepper. Refrigerate until ready to serve.

For the Duxelles: In a food processor, pulse mushrooms, shallot, and garlic until finely chopped, about 15 pulses, scraping down sides of bowl as needed.

In a 12-inch skillet, melt butter over medium-high heat. Add pulsed mushroom mixture, thyme, 1 teaspoon salt, and ¼ teaspoon pepper and cook, stirring often, until mushrooms are cooked and browned and nearly all moisture has evaporated, 6 to 8 minutes. Stir in Madeira, if using, and cook until no liquid remains, about 2 minutes. Transfer to a plate and let cool completely, about 20 minutes.

For the Salmon en Croûte: In a small bowl, whisk Dijon mustard and dill to combine; set aside. Line a rimmed baking sheet with parchment paper; set aside. With a rolling pin, roll both sheets of puff pastry into 11-to 12-inch squares on a lightly floured counter. Place 1 pastry square, centered on the prepared sheet, and brush with half of the mustard-dill mixture. Center the salmon lengthwise down the sheet of puff pastry. Brush top and sides of salmon with remaining mustard-dill mixture. Spread cooled mushroom mixture evenly over top of the salmon, gently patting it firmly in place.

Stretch remaining pastry square over fillet to meet bottom pastry square. Cup your hands around salmon, pressing gently to create a tight shape.

Pinch top and bottom pastry squares together to seal. Trim excess pastry to form a circle or rounded oval, leaving a 1-inch border around the fillet.

Gather excess dough trimmings and roll out excess dough. Using a ruler and sharp knife or pizza cutter, cut dough lengthwise into ¼-inch-wide strips that are the width of the wrapped salmon.

Brush top, sides, and ends of pastry-wrapped salmon with prepared egg wash; refrigerate remaining egg wash. Lay cut strips of dough diagonally across the top, spacing them evenly. Gently press strips to adhere and trim excess at each end if needed.

Crimp dough evenly around the edge using your fingers to make an attractive fluted border. Refrigerate pastry packet, uncovered, until cold, about 1 hour.

When ready to bake, adjust oven rack to upper-middle position and heat oven to 400°F (205°C). Brush tops and all sides of pastry with remaining beaten egg.

Bake until pastry is crisp and golden brown, and thickest part of the fillet registers 120°F to 130°F (50 °C to 54 °C) for medium, 20 to 30 minutes. Transfer sheet to wire rack and let salmon en croûte rest, 10 to 15 minutes. Slice and serve with dill sauce.