Season the fillets with salt and pepper
On a mandolin or with a very sharp knife, thinly slice the potatoes.
Set out one piece of parchment paper per fillet. Place overlapping slices of potato on the parchment paper and top with one piece of fish. Carefully fold the parchment over the fish from both sides to cover completely. place the packet in the refrigerator to chill for about 30 minutes.
Heat a large saute pan to medium-high heat. Add 2 tablespoons butter to the pan. Let melt and place fish top-side down in pan. Add more butter as necessary.
Once golden brown on the bottom, carefully flip the fish over. Pour in the wine followed by the garlic and thyme. Let the wine cook down by half.
Swirl in 2-3 tablespoons butter. Spoon the sauce over the fish fillets until fully cooked through.
Top with Parmeggiano-Reggiano cheese. Cover to let melt.