Make bechamel sauce: Cook onion in butter in a 2-quart heavy saucepan over moderately low heat, stirring, until softened.
Add flour and cook, stirring, 3 minutes.
Add milk in a stream, whisking, and bring to a boil, whisking.
Add salt and white pepper, then reduce heat and simmer, stirring occasionally, 10 minutes.
Pour béchamel sieve into a bowl and cover surface with a round of wax paper.
Make croquettes: Preheat oven to 350°F Tear bread into pieces and pulse in a food processor until reduced to fine crumbs.
Spread 3/4 cup bread crumbs in a shallow baking pan (reserve remaining bread crumbs for coating) and bake, stirring occasionally, until dry but not golden, about 5 minutes.
Heat 2 tablespoons butter in a 1-quart heavy saucepan over moderately high heat until foam subsides, then cook mushrooms and onion with lemon juice, 1/4 teaspoon salt, and 1/2 teaspoon pepper, covered, stirring occasionally, until mushrooms give off some liquid and onion is slightly softened, about 5 minutes.
Add scallops and enough water to just cover them, then bring to a boil over moderate heat.
Remove from heat and transfer mixture with a slotted spoon to processor, reserving liquid in pan.
Add cheese and parsley sprigs to processor and pulse until all of mixture is minced, then transfer to a large bowl.
Boil liquid in pan over high heat until reduced to about 2 tablespoons and stir together with 1/3 cup béchamel sauce, reserving leftover sauce for another use.
Add to scallops, stirring gently, then fold in dried bread crumbs.
Chill, covered, at least 4 hours.
To form croquettes: Put flour, egg, and remaining (fresh) bread crumbs in separate shallow bowls and stir remaining 1/4 teaspoons salt and pepper into crumbs.
Roll level tablespoons of scallop mixture into ovals.
Working in batches, coat ovals with flour, knocking off excess, then dip in egg, letting excess drip off, and roll in crumbs to coat.
ransfer to a wax paperlined baking pan as coated.
Cook croquettes: Cut remaining 2 sticks butter into 1-inch pieces and melt in a heavy saucepan over low heat.
Remove from heat and let stand 3 minutes.
Skim froth and slowly pour butter into a 10-inch heavy skillet, leaving milky solids in bottom of saucepan.
Discard milky solids.
Heat butter over moderate heat until hot but not smoking, then cook croquettes about 6 at a time, turning, until golden brown all over, about 3 minutes.
Transfer with a slotted spoon to paper towels to drain and serve warm.