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In a mixing bowl, season ground beef to taste with salt and pepper.
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Heat a small amount of oil in saute pan and cook onion until tender.
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In a small bowl, beat egg lightly; add milk, mixing well.
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Stir in breadcrumbs and allow to rest.
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Add onions and breadcrumbs to ground beef, working ingredients together with your hands.
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Divide meat into fourths, and make a patty; toss back and forth to force air out.
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Shape into burger shape.
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Heat small amount of oil in skillet and cook burgers; cover pan, and simmer on medium-high heat until burgers cook done; then remove lid, reduce heat to medium, and cook till water disappears.
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Remove from pan.
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Add 2/3 cup water, ketchup, sake, Worcestershire sauce, soy sauce, and butter to pan; bring to boil.
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Simmer till thickened.
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Spoon sauce over burgers on plate.