Preheat the oven to 425°F (220°C). Adjust the oven rack to the lower-middle position. Line a rimmed baking sheet with parchment paper.
Combine the potatoes, garlic, and oil in a microwave-safe bowl and season with salt and pepper. Microwave, uncovered, until the potatoes are just tender, stirring once or twice, 5 to 10 minutes.
Arrange the potatoes, overlapping the edges slightly, in 4 portions that resemble rectangles roughly 4 by 6 inches (10 by 15 cm).
Pat the fish dry with paper towels, season with salt and pepper, and place a fillet on top of each stack of potatoes, skinned side down. Top each fillet with an equal amount of butter, thyme sprigs, and lemon slices.
Roast until the fish flakes apart when gently prodded with the tip of a paring knife, 15 to 18 minutes. Slide a thin metal spatula underneath each stack of potatoes and fish, gently transfer the whole shebang to a plate, and serve immediately.