Preheat oven to 350 degrees F (175 degrees C). Arrange potatoes on a baking sheet.
Bake potatoes in the preheated oven until tender, about 1 hour.
Melt 2 tablespoons butter in small pot over medium heat. Add green onions; cook until soft, about 4 minutes. Stir in flour until well blended, about 1 minute. Pour in milk and 1/2 cup half-and-half gradually, stirring sauce constantly. Season with salt and cook until sauce is thickened, 10 to 15 minutes.
Cut a slice from the top of each baked potato and scoop the pulp into a bowl. Mash pulp until no lumps remain. Add 1/4 cup half-and-half and 1 tablespoon butter; blend together with an electric mixer. Stir in 1/2 cup Parmesan cheese and pepper until fully combined.
Fill baked potato skins with the mashed pulp mixture; make a well on top of each potato. Fill wells with onion sauce. Sprinkle 1/4 cup Parmesan cheese on top.
Turn on the broiler. Return stuffed potatoes to the oven and broil until golden brown; about 15 minutes.