Preheat the oven to 375 degrees. In a small bowl, combine shallot, thyme, and bread crumbs; season with salt and pepper. Add butter and toss to combine.
Place apple halves cut-side up in a baking dish large enough to hold them in a single layer. Sprinkle with sugar and top with bread-crumb mixture. Pour 1/4 cup cider around apples. Cover and bake for 30 minutes. Uncover and bake 20 minutes more or until apples are tender and crumbs are crispy. Remove from the oven.
Turn oven to broil with the rack 4 inches from the heat source. Pat trout fillets dry with paper towels. Season both sides with salt and pepper. Brush a rimmed baking sheet with 1 tablespoon of oil. Place trout skin-side up on the baking sheet; brush trout skin with remaining tablespoon oil. Broil 6 minutes, or until skin is golden brown and trout is cooked through. Reheat apples on the shelf underneath the trout so the crumbs don't burn for the last 1 to 2 minutes.
Meanwhile, in a small saucepan, whisk together remaining 3/4 cup cider, mustard, and capers until well combined. Cook over medium-high heat until reduced to a sauce consistency, 5 to 7 minutes; you should have about 8 tablespoons of liquid remaining. Place two trout fillets side by side on each of four dinner plates. Spoon juices around the fish, set two apple halves next to each pair of fillets, and garnish with fresh thyme.