Combine salt, pepper, thyme, and paprika, and rub on all sides of the short ribs.
Preheat braiser or dutch oven on the stovetop over medium-high heat. Once it’s hot, coat the pan in a little bit of olive oil. Sear the short ribs for 2 minutes per side. Remove them from the pan and set aside.
Add onions, celery, carrots, and garlic to the pan. Saute the veggies, stirring occasionally, until they’ve started to caramelize. Add red wine to deglaze the pan, using a wooden spoon to scrape the brown from the bottom of the pan.
Return the short ribs to the pan and add the beef stock. The liquid should cover about half of the short ribs. Depending on the size of the pan, you may not need all of the beef stock – you don’t want them to be covered up with too much liquid. Add some water or more stock if the short ribs aren’t covered by half.
Move the pan to the oven and cook with the lid on for about 2.5 hours, until the meat easily comes off of the bone. Cook for an additional 15-20 minutes uncovered, then remove from the oven and let the short ribs rest.
Transfer the pan juices to a small saucepan. Heat over high heat until it begins to boil.
Combine 1 tbsp cornstarch with about 2 tbsp cold water. Stream the cornstarch mixture into the saucepan with the pan juices, whisking constantly. Allow to boil for about a minute. Reduce to low to keep warm, and stir occasionally.
Shred the short ribs with a fork. Transfer the meat into an oven-safe pan. Pour gravy over the short ribs and stir to combine. Cover and transfer to the oven to keep warm while making the risotto. (If the oven is still warm from the short ribs cooking, you can just leave the heat off.)
Simmer chicken stock in a saucepan over medium heat, then reduce to low to keep warm.
Meanwhile, heat a saute pan over medium heat, and coat with olive oil once it’s hot. Cook mushrooms, shallots, and garlic until the shallots are translucent.
Add the rice and toast, stirring constantly, for about a minute. Add the wine to the pan. Bring it to a boil, then turn back to medium heat.
Once the wine is reduced by half, add a ladle full of the stock to the rice. Stir occasionally until the stock has been almost entirely absorbed by the rice, then add another ladle of stock. Repeat the process of stirring and adding another ladle of stock. Add the heavy cream at the end when there is only a little bit of stock left.
Remove from heat, stir in the parmesan cheese. Add salt to taste.
Serve short ribs over bed of risotto. Add carrots from the braising liquid if desired.