Pan roast the mushrooms: Heat 1 tablespoon olive oil in a large skillet over medium-high heat. Once it shimmers across the pan, add some mushrooms so that they loosely cover the bottom of the pan. (Do not overcrowd the pan-you will not use all of the mushrooms, yet.) Season the mushrooms with salt and pepper, then allow them to cook for approximately 2 minutes, turning them as they brown in the pan. Add some (but not all) of the shallots, butter and thyme to the pan and continue to cook until the mushrooms are tender, approximately 2 additional minutes. Remove the mushroom mixture to a paper towel lined plate, then wipe out the pan once cool enough to handle. Repeat the process, in batches, until all the mushrooms have been roasted.
Marinate the mushrooms: Heat 2 tablespoons olive oil in a large sauté pan over medium-low heat until it shimmers and slides easily across the pan. Add the leeks, carrots and celery, then season the mirepoix with salt and pepper. Cook, stirring occasionally, until the vegetable begin to soften, approximately 10 minutes. Add the garlic, bay leaf and coriander until the vegetables are completely soft, an additional 10 to 15 minutes. Add the pan-roasted mushrooms to the pan and season again with pepper. Then add the vinegar, chicken stock, thyme sprigs and parsley. Simmer for a minute or two to blend the flavors, then add 1/4 cup olive oil. Remove the skillet from heat and cool to room temperature, then cover and refrigerate for at least 2 hours. Prepare your sides while the mushrooms are marinating.
Prepare the vinaigrette for the watercress: In a medium bowl, add the 3 tablespoons olive oil and lemon juice, then season with salt and pepper. Set aside.
Prepare the potatoes: Place the potatoes in a large saucepan, then fill it with water to cover them completely. Season the water with salt, then bring to a boil and reduce heat to a simmer until the potatoes are tender, approximately 30 minutes. While the potatoes are simmering, combine the milk, garlic, bay leaf and thyme in a medium saucepan. Bring the mixture to a boil over medium-low heat. Once the milk has reached a boil, remove from heat, and cover so that the herbs steep in the milk for 10 minutes. Once finished, remove the garlic, bay leaf and thyme and discard. Once the potatoes are tender, drain the water and peel them while they're still hot. Cut the potatoes into manageable pieces, then pass them through a food mill at it's smallest setting over a large pot. Re-fire the milk over medium-low heat. Beat in the butter, one piece at a time then slowly add the milk until each addition is adsorbed before adding more until the puree is fine. Season with salt and pepper, if necessary.
Braise the salmon: Heat 2 tablespoons olive oil in a large skillet over medium-high heat until it slides easily across the pan. Season the salmon with salt and pepper, then add the fillets to the pan, seasoned side down. Cook the salmon for approximately 3 minutes until browned, adjusting the heat so the oil does not smoke, if necessary. Remove the fillets to a plate and wipe out the skillet. Return the skillet to the stovetop over medium heat. Add the chicken stock and reserved marinated mushroom mixture, then the salmon, cooked side up into the braising fluid. Simmer, basting periodically, until the salmon is cooked through, approximately 2 to 4 minutes. Remove the salmon to the plate, then stir butter into the mushrooms and season with salt and pepper.
Finish the dish: Whisk the vinaigrette and toss with the watercress bunch. To plate, spoon some purée onto the plate, then top with mushrooms. Stack the salmon fillet on top of the mushrooms and garnish with the watercress.