Heat mushrooms, 4 tablespoons butter, and two-thirds of the shallots in a medium saucepan over medium heat; cook until the mixture forms a loose paste, about 25 minutes.
Stir parsley and minced tarragon into mushroom mixture; season with salt and pepper.
Divide mixture among 4 cleaned scallop shells or shallow gratin dishes.
Bring vermouth, bay leaf, salt, and 3/4 cup water to a boil in a 4-quart saucepan over medium heat.
Slice the scallops sliced along the equator (middle). Add scallops to the liquid and poach until barely tender, about 1 minute.
Remove scallops from the heat. Place three slices over mushrooms in each of the shells.
Add the remaining shallots to the cooking liquid and boil until reduced to 1/2 cup, about 10 minutes; set aside.
Heat broiler to high.
Heat remaining butter in a small saucepan over medium heat. Add flour; cook until smooth, about 2 minutes.
Strain the reduced cooking liquid into the flour-butter mixture and add cream; cook until thickened, about 8 minutes.
Add cheese and lemon juice; season with salt, and pepper and divide the sauce over scallops.
Broil until browned on top, about 4-5 minutes; garnish each with a tarragon leaf.