2Season beef generously with salt. In a Dutch oven, brown beef in 4 tablespoons butter and 2 tablespoons olive oil over high heat until deeply crusted. Remove beef and set aside.
3In the same pot, add rough-chopped carrots, celery, and red onion. Cook until softened and lightly caramelized.
4Return the beef to the pot, then add the full bottle of red wine, beef broth, halved garlic heads, rosemary, and thyme. Cover and cook at 325°F (160°C) for 3 hours, until beef is fall-apart tender.
5While beef cooks, boil whole Yukon gold potatoes in heavily salted water until tender. Drain, let steam dry, then mash or rice the potatoes. Mix with milk, melted butter, and ½ cup Gruyère cheese until thick and sturdy.
6In a large cast-iron skillet, heat 3 tablespoons olive oil. Cook diced yellow onion, carrots, and celery over medium-low heat until very soft. Add minced garlic and tomato paste; cook until fragrant.
7Shred the cooked beef and add it to the skillet with the vegetable mixture.
8Strain the braising liquid and add it to the skillet. Bring to a low simmer. Mix 4 tablespoons butter and 4 tablespoons flour together, then stir into the skillet to thicken. Season to taste.
9Spoon mashed potatoes over the beef mixture in the skillet. Sprinkle remaining ½ cup Gruyère and ½ cup Parmesan cheese over the top.
10Bake at 400°F (200°C) for about 30 minutes, until golden and bubbling. Broil for a few minutes to brown the top. Let rest 5 minutes before serving.