In a medium skillet, heat the oil over medium-high heat. Season the salmon fillets evenly with the salt and place each fillet flesh-side down in the hot skillet. Cook until deep golden brown, about 3 minutes. Sear the sides of the fillets for 1 minute on each side. Using a fish spatula, flip the fillets so the skin side is down and cook an additional 2 minutes. Add 1 tablespoon of the butter and the oregano sprigs to the skillet and reduce the heat to medium. Baste the salmon fillets with the infused butter for 1 minute. Remove the salmon and oregano sprigs to a plate to rest.
In the same skillet over medium heat, add the artichoke hearts and shallots. Cook, stirring often with a wooden spoon, until the shallots are softened and the artichokes are beginning to brown, about 4 minutes. Add the chopped oregano and tomatoes and stir to combine. Deglaze the skillet with the white wine and allow the wine to cook down until reduced by half. Stir in the remaining 1 tablespoon butter and the spinach. Cook until the spinach is just barely wilted. Spoon the vegetables onto a platter and top with the rested salmon. Spoon any remaining sauce over the fish and serve.