Lay trout, skin side down, in a 9- by 13-inch baking pan; brush tops lightly with olive oil and sprinkle with salt and pepper. Bake in a 350° oven until opaque but still moist-looking in center of thickest part (cut to test), 7 to 8 minutes. Let cool. With a fork, lift flesh from skin and flake into a bowl. Add 1/3 cup bread crumbs, mayonnaise, and fresh tarragon; mix, and add salt and pepper to taste.
Pour vegetable oil into a 10- to 12-inch nonstick frying pan over medium-high heat. Put another 1/2 cup bread crumbs in a shallow bowl. Form trout mixture into six 2-inch-wide cakes, turn each in the bread crumbs to coat, and lay them slightly apart in pan. Cook, turning once, until golden brown on both sides, about 6 minutes total per cake.
Transfer to a paper towel-lined baking pan and keep warm in a 200° oven up to 20 minutes.