For the salsa verde: In a food processor, combine the olives, cocktail onions, cornichons, capers and anchovies and blend until smooth. Set aside in a bowl.
In a food processor or blender, puree the parsley with the olive oil until smooth. Do not puree too long or it may get too hot and cause the parsley to darken. Reserve in the food processor. Just before serving, pour the parsley oil into the olive-caper mixture and mix to combine.
For the leeks: In a saute pan over medium heat, add some olive oil. Add the leek coins and thyme and season with salt. Cook, stirring frequently, until the leeks start to wilt, 8 to 10 minutes. Deglaze with some vegetable stock and cover. Continue to cook, adding more stock as necessary, until the leeks are tender, another 10 to 15 minutes. Add additional salt if needed. Reserve.
For the branzino: Heat a large cast-iron pan over medium-high heat. Season the fish with salt.
Put some canola oil in the pan and wait until it's just about to hit the smoking point. Place the fish in the pan skin-side down (at this point the fish will buckle). Wait 30 seconds, then press the fish down so the skin makes contact with the pan and begins to crisp up. Reduce the heat to medium and continue to cook until the skin is crispy, 3 to 4 minutes. Place the thyme sprig on the fish and spoon some of the hot oil over the flesh to finish cooking it, about 1 minute. Remove to a paper towel-lined plate.
Add the mushrooms to the pan and cook, stirring, for 2 minutes. Add the leeks and clams and deglaze with some vegetable stock. Cover and cook until the clams steam open, about 2 minutes. Finish with the parsley and olive oil and adjust the seasoning.
Put 1 tablespoon of the salsa verde on a plate, arrange the vegetables and clams on top and top with the fish fillet.