For tapanata, using a food processor, pulse ingredients until a coarse paste forms, season with freshly ground black pepper and set aside.
For white polenta, bring stock and milk to the boil. Add polenta in a thin, steady stream, whisking continuously until incorporated. Cook over low heat, stirring occasionally with a wooden spoon to prevent polenta from sticking to base of pan, until smooth and creamy (about 15 minutes). Stir through butter and parmesan, and keep covered with baking paper to prevent a skin forming.
Meanwhile, preheat oven to 200C. Season trevalla with freshly ground black pepper, and spread tapanata on one side. Heat olive oil in a large ovenproof frying pan, add fish and cook, tapanata-side up, until sealed (2-3 minutes). Add butter to pan, transfer to oven and bake until cooked through (4-5 minutes). Divide polenta among serving bowls, top each with a trevalla fillet and serve immediately with pan juices, a drizzle of extra-virgin olive oil, and lemon halves to the side.
Note If Riviera olives are unavailable, substitute with any black olives.