In the same frying pan, melt remaining butter on medium heat. Add flour and stir until it turns a light golden color. Remove from heat; smoothly mix in broth, cream and vermouth. Return to high heat; bring to a boil, stirring and set aside.
Fill a 4 to 5-quart pan 3/4 full with water and bring to boiling over high heat. Add lasagna to water and boil, uncovered, until tender to bite, about 10 minutes; drain. Rinse lasagna with cold water until cool; drain again and set aside. Pour scallop juices into a 1 to 1 1/2-quart pan; boil on high heat, uncovered, until reduced to about 2 tablespoons; stir near the end to keep juices from scorching. Mix juices with cream sauce.
Line bottom of a 9 x 12-inch buttered baking pan or dish with 1/3 of the lasagna noodles. Spread noodles with 1/3 of the sauce; top with 1/3 of the scallops, then with 1/3 of the cheese. Repeat layers twice more, ending with cheese on top.
Cover pan with a lid or foil; if made ahead, chill up to 1 day, then bring to room temperature. Bake lasagna in a 350 degrees oven for 20 minutes. Remove foil and bake until cheese is golden, about 20 minutes longer. Let stand 15 minutes to firm before cutting; lift out portions with a wide spatula. Makes 12 servings as a first course, 6 as an entree.