In a large bowl, whisk the beer, 1 cup plus 2 tablespoons flour, the salt and paprika until smooth. Chill for 30 minutes.
In a medium bowl, combine the slaw mix and 1/3 cup Russian dressing.
Preheat the broiler. Meanwhile, in a deep skillet, heat the oil until it registers 375° on a deep-fat thermometer. Place the remaining 1/4 cup flour on a large plate. Working 1 at a time, coat the fish fillets with the flour, shaking off any excess, then coat with the beer batter. Lower the fish into the oil and cook, turning occasionally, until brown and crispy, about 5 minutes. Transfer to paper towels to drain.
Place the bread on a baking sheet; broil until toasted. Transfer toasted side down onto a work surface. Spread the remaining Russian dressing on 4 slices, then top with the fish, slaw, cheese and remaining bread slices, toasted side up.