For the thyme crème fraîche sauce:
Combine the stock, shallots, thyme sprigs, and pepper in a medium saucepan and bring to a boil over high heat. Reduce until thick and syrupy, about 40 minutes. Reduce the heat to medium, add the crème fraîche, and bring to a boil. Simmer until bubbly and slightly thickened, about 3 minutes. Strain through a fine strainer and season with salt (if needed) and pepper.
Stir the thyme leaves into the reduced stock mixture and keep warm.
For the leek strips and stuffing:
Cut each leek lengthwise through the outer 1 or 2 layers only and remove those layers. Flatten and cut lengthwise into ¼-inch-wide strips (you will need about 20). In a large pot of boiling salted water, blanch the strips for 30 seconds. Shock in a bowl of ice water and drain.
Cut the remaining white part of the leeks in half lengthwise and then slice crosswise into ¼-inch half-moons. In a small skillet, heat the butter over medium-high heat until bubbling. Add the sliced leeks and cook until tender, 3 to 4 minutes. Raise the heat to medium-high, add the stock and thyme leaves, and cook until the leeks are soft and the stock is reduced to less than a tablespoon. Season with salt and pepper, drain and cool at room temperature.
For the trout:
Preheat the oven to 350°F.
Lay 5 leek strips crosswise on a flat surface, spacing them every 1½ to 2 inches. Open a trout flat and season lightly with salt and pepper. Place over the strips lengthwise, and stuff with leek and thyme stuffing. Close the fish over the stuffing, and tie each strip around the fish, knotting it on top. Repeat with the remaining fish, leek strips, and stuffing.
Place the stuffed fish on a lightly oiled baking sheet and bake until medium-rare, 25 to 30 minutes.
With a sharp knife, slice the fish between the ties to make 3 medallions, each one wrapped with a leek strip, discarding the head and tail portions. Spoon some thyme sauce onto each plate and place 3 medallions on top. Or serve whole, accompanied by the sauce.
Substitute Red Wine Salmon Sauce or a flavored oil for the thyme crème fraîche sauce.
Substitute mushroom or small-diced vegetables for leeks in the stuffing.
Serve the stuffing as a foundation for any fish fillet or fish steak, and drizzle with any sauce or flavored oil.
Substitute coho salmon (or a larger fish, increasing the stuffing and the cooking time) for rainbow trout.