In a saucepan over medium heat, add the butter, oil, and onions. Stir together and cover with a lid. Cook until onions are softened and deep golden brown, stirring occasionally.
Stir in the flour and cook for 1 minute. Add the broth, thyme, salt and pepper and bring to a simmer. Add in the beef and simmer until the sauce concentrates in flavor. Once its is flavorful remove the thyme, season with salt and pepper.
Preheat the oven broiler to low.
Place the halved rolls on a foil-lined baking sheet. Top each roll with beef onion mixture and cheese.
Place under the broiler until the cheese is melted and golden brown.
Drizzle some of the broth over top or serve on the side. Serve with pickle and chips. Garnish with parsley.