"The combination of flavors is amazing! Richness from the braised beef, silkiness of homemade pasta, and then a little sharp bite from the arugula ties it all together. Shred beef or serve ribs whole."
Heat 2 tablespoons olive oil in a Dutch oven over medium-high heat until almost smoking. Season short ribs with salt and pepper. Cook in the hot oil, seasoned-side down, until browned and crusty, 5 to 6 minutes per side. Transfer to a plate.
Combine onion, carrots, celery, and garlic in the Dutch oven. Season with salt and pepper. Cook and stir until starting to soften, 8 to 10 minutes. Pour in red wine; scrape browned bits off the bottom of the Dutch oven using a wooden spoon. Pour in chicken stock. Bring to a boil; reduce heat and simmer to cook off alcohol, about 15 minutes.
Return short ribs to the Dutch oven, ensuring they are mostly covered with liquid. Cover.
Bake in the preheated oven until beef comes off the bone when pulled with a fork, 2 to 2 1/2 hours.
Bring a large pot of lightly salted water to a boil. Cook pasta at a boil until tender yet firm to the bite, 3 to 4 minutes. Drain. Toss with 1 tablespoon olive oil to prevent sticking.
Fill serving bowls with pasta. Scatter arugula on top. Spoon beef with juices over arugula. Garnish with Parmesan cheese and red pepper flakes.