Preheat oven to 350°F. Season short ribs liberally with salt and pepper. Heat oil in a large Le Creuset Dutch oven over medium-high heat. Working in 2 batches, brown short ribs on all sides, about 8 minutes per batch. Transfer short ribs to a plate. Pour off all but 2 tablespoons drippings from pot.
Add onions, carrots, and celery to pot and cook over medium-high heat, stirring often, until onions are browned, about 5 minutes. Add flour and tomato paste; cook, stirring constantly, until well combined and deep red, 2-3 minutes. Stir in wine, then add short ribs. Bring to a boil; lower heat to medium and simmer until wine is reduced by half, about 25 minutes. Add all herbs to pot along with garlic. Stir in stock. Bring to a boil, cover, and transfer to oven.
Cook until short ribs are tender, 2 - 2 1/2 hours. Transfer short ribs to a platter. Strain sauce from pot into a measuring cup. Spoon fat from surface of sauce and discard; season sauce to taste with salt and pepper.
While the short ribs are cooking, prep the potatoes. Peel and dice potatoes, making sure all are in about 1 inch cubes. Place in a large pot and cover with water. Bring to a boil over medium-high heat and then reduce heat to medium low to maintain a rolling boil. Cook until potatoes are fork tender and you can easily pierce them with a fork when touched.
Remove the potatoes from the heat and drain off the water. Place the diced potatoes through a ricer and then place back into the warm pot. Add the cream, butter, Parmesan, creme fraiche and stir to incorporate. Season with salt and pepper and fold in the chopped chives.
Combine the garnish ingredients together and toss to combine.
Serve in shallow bowls over mashed potatoes with sauce spooned over and gremolata on top.