Using a paper towel, dry the sole fillets very well. Season the fish evenly with 1 teaspoon salt. Heat a medium skillet over high heat. Add 1 tablespoon olive oil and 1 tablespoon butter to the pan. When the butter is fully melted and the bubbles have subsided, dredge both sides of 2 fillets in the flour. Shake off the excess flour and add the fish to the skillet. Reduce the heat to medium high. Cook the fillets until beginning to brown around the edges on the first side, 2 to 3 minutes. Using a fish spatula, flip the fish gently and cook for another 30 seconds. Remove the fillets to a plate and continue with the remaining 1 tablespoon olive oil, 1 tablespoon butter and fish.
When all 4 fillets are cooked and out of the skillet, add the capers and garlic and cook over medium heat, stirring, until fragrant, about 15 seconds. Add the chicken stock and lemon juice and stir, scraping up the bits from the bottom. Season with the remaining 1/4 teaspoon salt. Simmer for about 2 minutes to reduce the liquid slightly. Finish the sauce by stirring in the remaining 2 tablespoons butter, chili paste and oregano. Spoon the sauce over the fish, sprinkle with the parsley and serve.