Cooks Illustrated recommends a chuck-eye roast, which is what I used. I’ve found that it can be difficult to find though.
I added about 1/4 cup red wine with the broths.
1 boneless chuck roast (about 3½ pounds)
salt and pepper
2 tablespoons vegetable oil
1 medium onion, chopped
1 small carrot, chopped
1 small celery rib, chopped
2 cloves garlic, minced
2 teaspoons sugar
1 cup canned low sodium beef broth
1 cup low-sodium chicken stock
1 sprig fresh thyme
¼ cup dry red wine
1. Adjust an oven rack to the middle position and heat oven to 300F. Thoroughly pat the roast dry with paper towels; sprinkle generously with salt and pepper.
2. Heat the oil in a large heavy-bottomed Dutch oven over medium-high heat until shimmering but not smoking. Brown roast thoroughly on all sides, reducing heat if fat begins to smoke, 8-10 minutes. Transfer the roast to a large plate; set aside.
3. Reduce the heat to medium; add onions, carrots, and celery to pot and cook, stirring occasionally, until beginning to brown, 6-8 minutes. Add the garlic and sugar; cook until fragrant, about 30 seconds. Add the chicken and beef broths and thyme, scraping the bottom of the pan with a wooden spoon to loosen the browned bits. Return the roast and any accumulated juices to the pot; add enough water to come halfway up the sides of the roast. Bring the liquid to a simmer over medium heat, then cover tightly and transfer the pot to oven.
4. Cook, turning the roast every 30 minutes, until fully tender and a meat fork slips in and out of meat very easily (3½-4 hours). Transfer the roast to a carving board and tent with foil to keep warm.
5. Allow the liquid in the pot to settle about 5 minutes, then use a wide spoon to skim fat off the surface; discard thyme sprig. Boil over high heat until reduced to about 1½ cups, about 8 minutes. Add the red wine and reduce again to 1½ cups, about 2 minutes. Season to taste with salt and pepper.
6. Cut the meat into ½-inch slices, or pull apart in pieces; transfer the meat to a warmed serving platter and pour about ½ cup sauce over the meat. Serve, passing remaining sauce.