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Using a 2 or 3 inch biscuit cutter, cut 16 circles from the bread, and butter each circle.
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Drain the salmon and remove any skin and bone.
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Mash the fish with a fork; stir in the cream cheese and grated cheese.
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Season with salt and pepper to taste.
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Spread the mixture thickly on half the bread circles, top with remaining slices, and sandwich firmly together.
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Beat together the egg and milk.
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Season with salt, pepper, and paprika.
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Dip the sandwiches in the beaten egg and saute in hot oil until crisp and golden.
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Drain on paper towels and serve at once.