Combine the ground beef, parsley, scallion, salt and pepper. Divide evenly into four portions by first halving, then halving each half. Shape each into 3/4"-1" thick oval patties, about the size and thickness of hamburgers.
Place flour in a shallow dish. Dredge each patty and reserve any remaining flour.
Heat the olive oil and butter in a deep-sided cast iron skillet or sauté pan over medium-high heat. Add beef patties. Sauté 3 minutes on each side until browned. Do not cook all of the way through. Remove from pan to a plate and cover with aluminum to keep warm.
Add onions to the pan and sauté 5 minutes. Scrape up the fond or 'brown bits' in the bottom of the pan as the onions cook down. Sprinkle with sugar and sauté 1 minute more. Stir in garlic.
Make a well in the center of the pan amidst the onions. Add the tomato paste to the well and sauté until the paste begins to brown. Allow the tomato paste to brown and burn slightly. Keep moving the paste around the well using a metal spatula to keep it from burning completely.
Sprinkle onions with reserved flour and cook for 1 minute. Stir in the wine and scrape the fond. Add Kosher salt if desired. Add Herbs de Provence and stir, scraping the fond again. Add the beef stock. Reduce heat and simmer for 10 minutes while preparing the cheese toast.
To prepare the cheese toasts, preheat oven to 400°F and line a large baking sheet with parchment paper. Place the bread slices on a baking sheet.
Combine softened butter, garlic, and shredded Parmesan into a paste using the tines of a fork. Spread on one side of each slice of bread. Sprinkle the shredded Swiss evenly over the butter. Bake until bread is about halfway to crisp and cheese is bubbly, 8 to 10 minutes.
Return the hamburger steaks and all accumulated juices to the French onion gravy. Cover and simmer for 10 minutes. Remove from heat, uncover, and allow to stand for 5 to 10 minutes to allow the sauce to thicken.
Serve steaks atop cheese toasts with the red wine and onion sauce ladled over. Load all remaining French onion gravy into a gravy boat to be passed tableside. Garnish with parsley and freshly grated Parmesan cheese if desired.