Pull short ribs out at least 15 minutes before you intend to start cooking and season well on all sides with kosher salt and black pepper. Preheat oven to 350 degrees Fahrenheit.
In a large dutch oven, add 1 tablespoon of oil and heat on med-high for a minute or two. Sear about 1 minute per side on all sides of your short ribs, until they're nice and brown. You may need to do this in two batches to not crowd the pan. If you don't give them enough space, they won't brown properly.
Remove short ribs from dutch oven and set aside. Pour out any excess oil so only around 1 tablespoon remains. Add shallots and carrots, sautéing for 1-2 minutes, till soft and fragrant. Then add garlic, brown sugar, and thyme.
Return the short ribs to the pot and pour in apple juice and broth. Bring everything to a boil and cover with the lid. Place in the oven at 350F degrees for 2 to 2½ hours. These can also be left a little longer if needed, or until the meat is tender and falling off the bone.
Pull the short ribs out of the pot and skim any excess fat off the top of the leftover sauce. Simmer the sauce for 5-10 minutes until thickened, if it hasn't already thickened up. Return the short ribs to the pot and coat with sauce. Taste and adjust any seasonings.
FOR THE POLENTA: 10 minutes before you pull out the short ribs, bring the milk and broth for the polenta to a boil in a medium pan. Slowly whisk in the polenta a little bit at a time and continue whisking as it slowly thickens - about 30 minutes. Whisk consistently - so pull up a stool or chair if needed. It's important to find the right temperature (usually medium low heat) for a mild simmer while whisking (small bubbles on the outsides). It should NOT boil. Keep warm until ready to serve. If it gets too thick, you can add a little water, broth, or milk to thin out the polenta. You don't want the polenta to cook too quickly or it will not soften properly.
Stir the mascarpone and butter into the polenta and season with salt and pepper to taste. Serve the short ribs on top of the polenta with a little sauce.