In a shallow bowl, whisk eggs, flour, mustard, garlic powder, ½ teaspoon salt, and ¼ teaspoon pepper. Place panko in another shallow bowl. Season fish all over with salt and pepper.
Into a large heavy skillet, pour oil to a depth of ¼". Heat over medium-heat until shimmering. Coat each piece of fish with egg mixture, then dip into panko, gently pressing to adhere.
Fry fish, turning halfway through, until golden brown and just cooked through, 2 to 3 minutes per side. Transfer to a paper towel-lined plate; season with salt.
Top bottom buns with slice of American cheese, then fried fish. Spread tartar sauce on top of the fish. Close sandwiches and serve with more tartar sauce alongside.
