Pull short ribs out at least 10 minutes before you intend to start cooking and season them with kosher salt and pepper. Preheat oven to 350.
In a heavy 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 so you don't crowd your pan. If you put too many in at one time, they won't brown nicely.
Remove short ribs from dutch oven and set aside. Pour out any excess oil so only around 1 tablespoon remains. Add shallots and carrots and saute for a minute or two, till soft. Then add garlic, brown sugar, and thyme.
Return the short ribs to the pot and pour in apple juice and stock/broth. Bring everything to a boil and then cover with a sturdy lid or tight foil and place in the oven at 350 degrees for 2 to 2 ½ hours. These can also be left longer if needed. It's tough to ruin short ribs.
FOR THE POLENTA: 10 minutes before you pull out the short ribs, bring the milk and broth/stock 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 15-20 minutes. You want it to be a mild simmer while you whisk, not a boil, so you may need to adjust your heat accordingly. Keep warm until ready to serve. If it gets too thick, you can add a little water, broth, or milk to thin out.
Pull the short ribs out of the pot and skim any excess fat off the top of the liquid mixture. Simmer the liquid 5-10 minutes until thickened if it hasn't already thickened up. Then return the short ribs to the pot and coat with sauce.
Stir the mascarpone and butter into the polenta and season with salt and pepper. Serve the short ribs on top of the polenta with a little sauce.