Preheat the oven to 425°F with a rack in the center position. Line a sheet pan with parchment paper.

In a large bowl, toss the potatoes with the olive oil, garlic, salt, and pepper until thoroughly coated.

Assemble 4 potato rafts by overlapping potato slices on the prepared pan in rectangular mounds. Each raft should consist of 3 or 4 shingled rows and be roughly 4 x 6 inches; use 3 or 4 slices of potato per row.

Roast the potatoes, rotating the pan halfway through, until golden brown and beginning to crisp, about 30 minutes. Remove the pan from the oven.

Blot the fish fillets dry with a paper towel. Place one, skin-side down, centered on top of each potato raft. Top each piece of fish with 1 tablespoon butter, 2 lemon slices, and 2 sprigs thyme. Scatter the capers atop the fish and around the pan.

Return the pan to the oven and roast until the fish is flaky and opaque, about 15 minutes.

Transfer the potato rafts and accompanying fillets to individual plates, ideally with a big spatula. Serve hot.