Preheat the oven to 400 degrees F.
Fold both sheets of parchment paper in half, making them large enough to hold the fish with a few inches of margin, when folded in half like a book. Aggressively season the fish inside and out with salt and pepper. Place 1 fish inside of each "book". Sprinkle each fish with the garlic slices, chile flakes, thyme leaves, olive oil, lemon juice, and wine. Seal the edges of the parchment together with the egg white and fold to form a closed compartment. Place the packages on a sheet tray and bake in the oven 20 minutes. Remove, slash the package to release steam and serve immediately, 1 fish for 2 servings. Drizzle with good oil and serve with lemon wedges.