Preheat the oven to 400 degrees F. Toss the potatoes with 1 tablespoon olive oil, 1 teaspoon thyme, 1/4 teaspoon salt, and pepper to taste in a bowl. Coat a rimmed baking sheet with cooking spray; add the potatoes and spread in an even layer. Roast until browned and crisp, tossing halfway through, about 35 minutes.
Meanwhile, toss the tomatoes, olives, capers and garlic with 1 teaspoon olive oil, 2 tablespoons vinegar and 1 teaspoon thyme in a bowl. Coat another baking sheet with cooking spray; add the tomato mixture and spread in an even layer. Roast until softened, 15 minutes.
Mix the remaining 2 teaspoons olive oil and 1 teaspoon thyme, 2 tablespoons vinegar, 1/4 teaspoon salt, and pepper to taste in a bowl; brush on the fish. Place the fish on top of the roasted tomato mixture and return to the oven until just cooked through, about 10 minutes. Divide among plates and serve with the potatoes.