Preheat the oven to 425℉. In a medium bowl, toss the potatoes with the sliced lemon, thyme leaves, and 2 tablespoons of the olive oil. Season with salt and pepper. On a rimmed baking sheet, shingle the potatoes and sliced lemon into four 6-by-4-inch rectangular piles. Roast until just tender and beginning to brown, about 30 minutes.
Meanwhile, prepare the fish. In a medium microwave-safe bowl, toss the panko breadcrumbs with the remaining 2 tablespoons of olive oil. Season with salt and pepper. Microwave on HIGH, stirring once, until toasty and beginning to brown, about 90 seconds; let cool.
Spread the flour in a shallow bowl and season with salt and pepper. In a separate shallow bowl, lightly beat the eggs.
Pat the cod dry with paper towels and season with salt and pepper. Working one at a time, dredge the fish in the flour, then dip in the egg, letting the excess drip back into the bowl. Then dredge the fish in the panko, pressing lightly to help it adhere. Transfer to a plate.
Remove the potatoes from the oven and lay one cod fillet on each pile of potatoes. Bake until the cod is crispy and just cooked through, about 15 minutes more.
Meanwhile, make the tartar sauce. In a medium bowl, whisk the mayonnaise with the capers, dill, parsley, the 2 tablespoons of lemon juice, and ¼ teaspoon of pepper.
Remove the baking sheet from the oven. Slide a spatula underneath the potatoes and cod and gently transfer to four plates. Garnish with more dill and parsley and serve right away, passing the tartar sauce and lemon wedges at the table.