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For the Short Rib Ragù:
Pat the short ribs dry and season generously with salt.
Heat a deep pot or Dutch oven over medium-high heat with oil.
Sear the short ribs on all sides until deeply golden, working in batches.
Remove and set aside.
Discard the darkened oil and add a thin fresh layer.
Lower heat to medium and add onion, carrot, and celery with a pinch of salt.
Sweat for a few minutes until softened but not browned.
Push vegetables to one side and add garlic directly to the pan.
Sauté until fragrant, then stir into the vegetables.
For the Fresh Pappardelle:
Mound the flour on a clean work surface.
Use the bottom of a bowl to create a wide well.
Add eggs, yolks, and salt to the center.
Whisk eggs until fully combined.
Gradually incorporate flour with a fork until a thick paste forms.
Use a bench scraper to fold remaining flour over the egg mixture.
Chop and fold until a shaggy dough forms.
If dry, lightly wet your hands and continue kneading.
If wet, add small pinches of flour until it just holds together.
For the Final Assembly:
Add a few ladles of ragù to a wide pan over medium heat.
Add just-cooked pasta and a ladle of pasta water.
Simmer until the sauce lightly coats the noodles.
Grate in additional Parmigiano to enrich and tighten the sauce.
Transfer to plates.
Finish with more Parmigiano and freshly cracked black pepper.