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1
Heat a large skillet, preferably cast iron, over medium-high. Add butter and oil and swirl pan until butter is melted. Add mushrooms, thyme, and garlic and season with salt and pepper. Cook, tossing often, until mushrooms are golden brown and tender, 5-8 minutes; set aside.
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2
Prepare a grill for medium-low heat. Using a sharp knife, make a slit along belly of salmon, running from rib cage to tail. Turn salmon on its back and cut along both sides of backbone without piercing through skin on other side. Carefully remove backbone without tearing skin, then remove pin bones with small needle-nose pliers or tweezers. Season inside of fish with salt and pepper and stuff lemon slices, dill, thyme, and tarragon inside cavity. Tie wet kitchen twine around fish in several places to secure. Place on a rimmed baking sheet; rub skin all over with oil.
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3
Carefully place fish on grill grates and grill, uncovered, until flesh on bottom side is mostly opaque but still slightly translucent in the center, 12-18 minutes. Carefully turn fish and cook on the other side to the same doneness, 8-10 minutes. Transfer fish to a clean rimmed baking sheet and let rest at least 10 minutes.
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4
Place skillet with chanterelles directly on edge of grill to reheat while you portion the fish. Remove strings and peel back skin from salmon; discard. Break flesh of top fillet into large pieces and transfer to a platter. Discard aromatics and remove skin from bottom fillet, break up, and transfer to platter. Season with more salt and pepper and spoon warm chanterelles over.
