In a large Dutch oven, heat the oil over medium-high heat. Sprinkle the short ribs with 2 teaspoons of the salt. Add the ribs to the pan (working in batches as needed) and sear each side until browned, 3 to 5 minutes per side, 12 to 15 minutes total. Remove the short ribs to a bowl, cover with foil to keep them warm, and set aside.

Add the onion, carrots, celery, and garlic to the pot. Sauté, stirring occasionally, until the onion is translucent, about 5 minutes.

Add the wine to the pot, scraping any browned bits from the bottom of the pan. Cook until the wine is reduced to ¼ cup, 4 to 5 minutes. Add the beef broth and thyme, bring to a low boil, and reduce the heat to a simmer.

Return the short ribs to the pot, with any juices from the bowl. Cover and simmer until fork tender, 4½ to 5 hours.

Remove the ribs to a bowl and tent with foil to keep warm. Strain the liquid.

Pour the liquid into a medium saucepan and simmer over high heat until slightly thickened and reduced to 1 cup, 8 to 10 minutes. Stir in the remaining 1 teaspoon salt and the pepper.

Spoon the sauce over the ribs and sprinkle with the parsley. Serve with gnocchi.

Store in an airtight container in the refrigerator for up to 3 days.

Carefully halve the potatoes lengthwise and scoop the insides into a bowl to cool, about 15 minutes. Press the insides of the potatoes through a potato ricer into a large bowl. Set aside until cool enough to work with.

Create a well on a flat surface. Crack eggs into the center and mix with the potatoes. Working quickly, sprinkle the flour and the remaining 1 teaspoon salt over the potatoes and gently fold in the flour and salt. Turn the dough out onto a lightly floured surface and gently knead just until the dough comes together. Divide the dough into 8 equal portions and cover with a clean kitchen towel.

Working with one portion of dough at a time, use your hands to roll the dough back and forth while moving them outward to elongate the dough into a ¾-inch-diameter rope. Using a dough scraper or knife, cut the rope into 1- to 2-inch pieces.

Bring a large pot of salted water to a boil over high heat. Line a sheet pan with a clean kitchen towel. Working in 3 or 4 batches (so the gnocchi have plenty of room to cook), gently add the gnocchi to the simmering water. (Be sure to let the water return to a simmer between batches.) When they rise to the surface, cook until slightly firm, about 1 minute. Using a spider or slotted spoon, transfer the gnocchi to the towel-lined sheet pan. Let cool for 10 minutes.

In a large nonstick skillet, heat 1 tablespoon of the canola oil over medium-high heat. Add half of the gnocchi. Cook, stirring occasionally, until golden brown, 4 to 6 minutes. Transfer to a plate and repeat with the remaining canola oil and gnocchi. Return the browned gnocchi to the skillet and toss with the sauce of your choice.