Sprinkle both sides of the halibut with the salt, pepper and granulated garlic. Compress the hash browns between 2 paper towels until they are flat and about the size of all 4 fillets. Eyeball the size and shape of the fillets and cut out hash brown "hats" to fit the top of each fillet. Dip the presentation side of each piece of halibut into the egg, then press a hash brown "hat" onto the egg.
Put the melted butter and oil into a cold, 12-inch nonstick pan. Place the fillets hash brown-side down directly into the butter and oil. Place on a burner and turn the heat to medium low. Cook, undisturbed, until the hash browns are golden brown, 8 to 10 minutes. Gently flip using a fish spatula and finish cooking until the fish registers 125 degrees F, about 5 more minutes. Serve immediately with the Mango-Mint Relish.
Mix together the mint, lime zest and juice, mango, chile and shallot in a bowl; season with salt and pepper.