Bring a large pot of heavily salted water to a boil. Make an ice bath with salted water.
Blanch the haricots verts in the boiling water until tender but still crisp, 3 to 4 minutes. Immediately shock in the ice bath to stop cooking. Drain the excess water and set aside on a sheet tray.
Coat a small saute pan with olive oil and add 1 tablespoon of the shallots and some salt (save the remaining shallots for the sauce). Sweat over medium heat until softened and translucent, about 1 minute, then add the mushrooms. Stir and cook until the mushrooms start to soften, 2 minutes more. Add a pinch of salt and the crushed red pepper flakes and continue cooking until golden brown, 2 to 3 minutes. Add the haricots verts to the pan and toss to coat. Taste for seasoning and set aside.
Put the flour in a shallow dish. Coat a large skillet with olive oil and heat over medium-high until the skillet is hot. Sprinkle both sides of the fish with salt and dredge in the flour. Gently shake off the excess flour and lay the fish skin-side down in the oil. Cook until the outside edges of the first side turn white and crisp, 2 to 3 minutes, then flip and cook until the fish turns white and opaque, 1 minute more. Remove the fish from the skillet and set aside on plates or a platter.
Use a paper towel to carefully wipe out the skillet used to cook the fish. Lower the heat to medium and lightly coat with olive oil. Add the remaining diced shallot with a pinch of salt and sweat until the shallots soften, about 30 seconds. Add the white wine and chopped lemon supremes and cook, stirring often, until it comes to a boil. Add the capers, lemon juice and lemon zest and simmer until the liquid is reduced by half. Add the chicken stock and a pinch of salt and continue to reduce at a simmer by half.
Add a pat of butter and cook, swirling the pan, until melted, then add another pat. As the butter melts, swirl the pan to fully incorporate. Continue adding butter, one pat at a time, until the sauce is thickened. (You should need between 3 and 4 pats total.) If the sauce reduces too much, adjust with a small splash of chicken stock to thin it out. Season to taste with salt and stir in the parsley.
Arrange the haricots verts and mushrooms next to the fish on plates. Top the fish with the sauce and garnish with a sprinkle of fresh parsley.
To cut supremes, peel and segment the lemon with a paring knife, separating the fruit from the membranes.