For the court bouillon: Add the mushrooms, wine, bay leaves, thyme, onions and 1 quart water to a saucepan. Bring to a simmer, then reduce the heat to medium-low and continue simmering about 1 hour. Strain.
For the salmon: Place a flameproof roasting pan over medium-high heat. Add the court bouillon and bring to a simmer. Add the salmon, skin-side up. Turn heat to low and bring to a slow simmer. Cook for 5 minutes per inch, measuring the thickest part of the salmon, about 10 to 15 minutes total. The salmon should be cooked to rare (a thin knife can be inserted into the center with slight resistance); keep in mind the salmon will cook a second time later in the recipe. Remove from the poaching liquid and let cool slightly. Break the salmon up into chunks and place in a bowl. Discard the skin and set aside.
For the mushroom duxelles: Add the shallots to a food processor and pulse to chop. Add the mushrooms and continue pulsing until a rough puree forms.
Add the butter to a medium skillet over medium heat and cook until foaming. Add the shallot and mushroom puree and cook until the mushrooms have released their liquid, 6 to 7 minutes. Add the cream and continue cooking until incorporated. Season to taste with salt and pepper. Set aside.
For the spinach: Add the butter to a medium skillet over medium heat until foaming, then add the onions. Saute until translucent, 4 to 5 minutes. Add the spinach and toss and saute until just wilted, 1 to 2 minutes. Season to taste with salt and pepper. Set aside.
For the crepes: Whisk together the eggs, butter and heavy cream in a large bowl. Whisk in both flours until incorporated and a pourable batter forms. Let the batter rest at least 10 minutes and up to 1 hour.
Heat a 9-inch nonstick crepe pan over medium heat, then add a thin layer of batter (about 1/2 cup). Spread around the pan to completely cover the bottom. Cook until the top is dry and the edges begin to curl up, around 1 minute. Flip and cook another minute, then remove the crepe to a sheet pan and repeat to make additional crepes, using all the batter.
To assemble: Preheat the oven to 400 degrees F.
Line a large roasting pan or rimmed baking sheet with parchment paper. Lay a thawed sheet of puff pastry on the bottom. Cover the puff pastry with 3 crepes, overlapping a bit. Add half of the cooked spinach followed by half of the salmon, leaving a slight border of puff pastry and crepes around the edges. Make a furrow in the middle of the salmon running the length of the pan, then add the duxelles. Top with the remaining salmon and spinach, in even layers. Cover everything with more crepes, overlapping a bit. Top with the remaining puff pastry sheet. Crimp the edges together to seal. Whisk the egg together with a tablespoon of water in a small bowl. Brush over the top. Brush the melted butter over the egg wash. Using a sharp knife, score the top in a cross-hatch pattern to release steam as it bakes. Bake until golden brown, about 45 minutes.
For the beurre blanc: Cook the white wine, champagne vinegar and shallots in a heavy nonreactive saucepan over medium heat to a glaze, about 10 minutes. Add the cream and bring to a simmer. Add the butter 1 tablespoon at a time, whisking until each piece is absorbed before adding the next piece. Season with salt and pepper. Reduce the heat to a bare simmer and keep warm.
For the rice pilaf: Add the butter, onion, a good pinch of salt and a few grinds of black pepper to a saucepan over medium heat. Add the wild rice and cook for 5 minutes. Add 2 cups cold water and bring to boil. Cover, reduce to a simmer and cook for 25 minutes. Let cool in the pan.
For the creme fraiche: Mix the creme fraiche and caviar together in a small bowl. Set aside.
Slice the coulibiac into 2- to 3-inch pieces. Drizzle with the beurre blanc. Serve with the creme fraiche and rice in bowls on the side.