Heat 2 tablespoons oil in a large Dutch oven over medium heat. Sprinkle the short ribs all over with 1 teaspoon salt. Brown the short ribs all over, in batches if needed, removing to a plate as they brown, 4 to 5 minutes.
When all of the ribs are out, add the carrot, celery and onion. Cook, stirring occasionally, until slightly softened, 3 to 4 minutes. Make a spot in the center of the pot and add the tomato paste, rosemary, thyme, red pepper flakes and three-quarters of the garlic. Let sizzle for a minute or so, then stir into the vegetables. Add the wine and bring to a boil, then cook until reduced by half, 1 to 2 minutes. Add the bay leaves and 3 cups stock. Return to a simmer and add the short ribs in one layer. Cover and bake, stirring halfway through and adding more stock to keep the ribs almost covered, until the short ribs are very tender and falling from the bone, 1 1/2 to 2 hours. Remove the short ribs from the oven. Use 2 forks to shred the meat, discarding the bones and cartilage.
For the crispy panko topping, heat the remaining tablespoon olive oil in a small skillet over medium heat. Add the remaining garlic and cook until sizzling. Add the panko. Cook and stir until the crumbs are coated with the oil. Continue to cook and toss until the crumbs are crisp and golden, about 3 minutes. Remove from the heat, then stir in 2 tablespoons parsley and season with salt. Set aside.
Bring a large pot of salted water to boil for the Gnocchi. Return the sauce in the Dutch oven to a simmer. Add the Gnocchi to the boiling water and boil until they float. Simmer until cooked through, 2 to 3 minutes from the time they begin to float. Remove with a large spider to the simmering sauce. Sprinkle with the remaining 2 tablespoons parsley and toss to coat the Gnocchi in the sauce, adding a little of the Gnocchi cooking water if the sauce seems too dry. Serve in pasta bowls, sprinkled with the crispy panko topping.
Dust a baking sheet with flour and set aside. Put the whole, unpeeled potatoes in a pot and cover with cold water by about an inch. Bring to a simmer and cook until tender when pierced with a knife, 15 to 20 minutes. Drain and let cool just enough that you can peel them. (They should still be quite warm.) Use a ricer or food mill to rice the potatoes right onto your work surface in a thin layer. Let cool completely.
Sprinkle the salt and some pepper over the cooled potatoes. Drizzle the beaten egg over the potatoes and sprinkle the flour over the egg. Flour your hands and knead the dough until it comes together in a loose clump. Knead just until the dough comes together in a cohesive mass (don't overwork it)—the dough should be a tight ball, not crumbly, but not too moist. You know the dough is right if you cut it in half and it's the texture of chocolate chip cookie dough (without the chips!).
Cut the dough into quarters, keeping the others covered as you work. Roll a piece of dough into a long snake about 3/4-inch thick and cut crosswise into 3/4-inch-thick pieces. Dust the pieces well with flour. Hold a fork in your non-dominant hand. Rest a piece of dough at the top of the tines of the fork and roll down with the thumb of your other hand, making a little concave depression in the gnocchi to catch sauce. Repeat with the remaining dough. Transfer the floured gnocchi to the prepared baking sheet. (Gnocchi can also be frozen right on the baking sheet—once they are solid, transfer to a zip-top bag for longer storage. Add a minute or two to the cooking time if cooking from frozen.)
You could substitute grated Parmesan for the panko topping if you're not keeping kosher.