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1
Season the short ribs with salt and pepper. Dredge the ribs in the flour, shaking off the excess.
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2
In a cocotte or Dutch oven over medium-high heat, warm the 1 Tbs. olive oil. Working in batches, brown the ribs on all sides, 6 to 8 minutes per batch, adding more oil to the pot as needed. Transfer to a platter.
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3
Add the onions and carrots to the pot and cook, stirring to scrape up the browned bits, until the onions are slightly translucent and beginning to brown, 4 to 6 minutes. Add the garlic and cook until fragrant, about 1 minute. Pour in the wine and simmer, stirring, until most of the liquid has evaporated. Add the demi-glace and tomatoes and cook, stirring constantly, for 2 minutes. Add the chicken and beef stocks, bay leaf and thyme sprigs, increase the heat to high and bring to a simmer.
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4
Return the short ribs and any accumulated juices to the pot. Cover, transfer to the oven and bake for 2 hours. Uncover and continue to bake until the beef is fork-tender and falling off the bone, 1 1/2 to 2 hours more. Discard the bay leaf and thyme sprigs.
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5
Using a slotted spoon, transfer the short ribs to a cutting board and let cool slightly. Remove the meat from the bones and cut the meat into 1-inch pieces. Return the meat to the pot and season with salt and pepper.
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6
Meanwhile, in a large saucepan over high heat, bring 4 cups salted water to a boil. Gradually whisk in the polenta. Reduce the heat to medium-low and simmer, uncovered, stirring occasionally, until the polenta is tender, thick and creamy, 20 to 25 minutes. If the polenta becomes too dry, add more water as needed. Whisk in the cheese and the 2 Tbs. butter.
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7
Spoon the polenta on top of the short rib mixture and spread evenly. Brush with the 1 Tbs. melted butter. Transfer to the oven and bake, uncovered, until the mixture is bubbling at the edges, about 30 minutes. Set the oven to broil, place the pot in the upper half of the oven and broil until the polenta is golden, 5 to 7 minutes.
