Melt 4 tablespoons butter in saucepan.
Saute the shallots for 1 minute.
Add the scallops.
Stir around.
Add vermouth and 1/2 cup water.
Simmer 5 minutes or until scallops have an opaque look.
Remove scallops.
Reduce juices to 3/4 cup.
Meanwhile, melt 3 tablespoons butter in skillet and add mushrooms, salt, pepper, and lemon juice.
Add spinach.
Toss mushrooms and spinach for 2-3 minutes.
Add juices, if any, to reduction.
Next, melt 2 tablespoons butter, stir in flour, and cook 1 minute.
Add the scallop juice reduction; whisk until thick.
With a fork, beat the egg yolk and heavy cream together.
Warm with a bit of the sauce and whisk this mixture into the sauce.
If too thick, thin with more cream, just to napping consistency.
Combine scallops and mushrooms and spinach mixture with cream mixture.
Place in an oval gratin dish.
Sprinkle top with bread crumbs and cheese and drizzle melted butter over.
Bake at 350° F until bubbly, about 15 minutes.
Place under broiler to brown, if desired.