Step 1Heat the oven to 375 degrees.
Step 2Cook the shallots in 2 tablespoons of the butter in a large pan over medium heat until translucent, about 3 minutes. Add the mushrooms and continue to cook another 5 to 10 minutes.
Step 3Add the spinach, kosher salt, pepper and nutmeg; continue cooking until the spinach is wilted, about 2 minutes. Transfer the mixture to a colander to drain; cool completely.
Step 4Place a small amount of filling inside the cavity of each fish. Combine the flour and salt on a flat plate, put the eggs into a shallow dish or pie pan. Combine the bread crumbs, tarragon, thyme and white pepper on another flat plate.
Step 5Dredge each stuffed trout in flour, shaking off the excess, then dip it in the eggs and then into the bread crumb mixture, pressing the crumbs onto the trout so that they will stay.
Step 6Bake the fish until it flakes with a fork, about 8 to 10 minutes.
Step 7While the fish is baking, melt the remaining 4 tablespoons butter in a small pan over low heat. Add the chopped nuts and cook the butter until it's golden brown.
Step 8To serve, place the trout on a serving platter and drizzle with the brown butter and pecan mixture.