Clean mushrooms with a damp towel and slice; set aside. Heat oil in a large sauté pan over medium-high heat. Add butter, thyme and garlic. Stir carefully with a wooden spoon so that the butter doesn't burn; if it starts to burn lower heat immediately. When butter is melted, add mushrooms and sear until golden brown. Season at the end with salt and pepper.
Heat oil in a large pot over medium-high heat; add onion, and garlic; stir and add kale stems and cook, stirring occasionally, about 5 minutes. Add kale leaves, vinegar, and 1 pint water. Add salt and pepper and cook, stirring occasionally, until liquid is mostly evaporated. Hold warm until ready to eat.
Place a heavy-bottom sauté pan or cast iron pan on high heat for 5 minutes. Pat each fillet dry with a paper towel. Season both sides lightly with salt and pepper. Add oil to the pan, add fish, skin-side down and sear. Reduce heat if oil starts to burn, but you want heat high enough so fish will sear properly. Let cook about 4 minutes. Flip to other side, turn off heat and let fish cook in pan another 2 minutes or until cooked to preferred doneness, being careful not to overcook. Remove fish to drain on paper towels.
Place kale and mushrooms onto two plates; top with fish and drizzle with some high quality extra virgin olive oil and a squeeze of fresh lemon juice.