Bring a large pot of salted water to a boil.
Heat 3 tablespoons oil in a large saute pan over medium-high heat. Add the garlic and onions and cook about 5 minutes. Stir in the tomato paste and red chile flakes. After a minute or so, add the chicken stock and pork and season with salt and pepper. Reduce the heat to low, cover and cook an additional 10 minutes.
When you are ready to serve, drop the pasta into the boiling water and cook 6 to 8 minutes. Remove the pasta and place directly into the sauce. Cook, tossing the pasta with the sauce, another 2 minutes, and then turn off the heat.
Place the pasta into warmed bowls, spooning extra sauce over the top of each portion. Garnish with Parmesan, parsley and chile flakes. Drizzle with olive oil, if desired.
Preheat the oven to 425 degrees F.
In a small bowl, mix the olive oil, garlic, salt, and pepper. Using a pastry brush, spread the mixture all over the pork shoulder.
Set the meat on a rack set into a roasting pan. Roast for 20 minutes, and then reduce the heat to 325 degrees F. Continue to cook until an instant-read thermometer inserted into the shoulder reads 185 degrees F, about 4 hours. Remove the pork from the oven and let stand until cool enough to handle, about 30 minutes.