Season beef liberally with salt and pepper, brown in a searing hot Dutch oven with butter and oil, remove.
Add in chopped carrots, celery, onion in same pot for 3-4 minutes. Return beef, add wine, broth, garlic heads, and tied herbs. Cover, cook at 325°F for approximately 3 hours until meat falls apart.
Boil potatoes until tender, drain, steam dry with tea towels for 5 minutes. Run them through a ricer or mash them, then fold in milk, butter, and ½ cup Gruyère.
Heat oil in 17" skillet, cook diced onion, carrots, celery until soft. Add garlic and tomato paste, mix and combine. Turn off heat.
Shred beef by hand once cooled, add to skillet, turn heat back on low.
Strain braising liquid into pan, bring to low simmer, make a slurry with butter and flour, stir in and season to thicken.
Spoon mashed potatoes over beef using an ice cream scooper for presentation and even distribution, then sprinkle remaining Gruyère and Parmesan across the top.
Bake at 400°F for approximately 30 minutes, broil a few minutes to brown top while keeping a close eye on it. Rest 5 minutes before serving.
