1 boneless chuck-eye roast (about 4 pounds)
Salt and pepper
1 tablespoon vegetable oil
1 onion, chopped medium
1 carrot, chopped medium
1 celery rib, chopped medium
2 garlic cloves, minced
2 teaspoons sugar
1 cup canned chicken broth
1 cup canned beef broth
1 sprig fresh thyme
1-1/2 cups water
1/4 cup dry red wine
Heat oven to 300 degrees.
Thoroughly pat roast dry with paper towels; sprinkle generously with salt and pepper. Heat oil in large heavy-bottomed Dutch (I use cast iron) oven over medium-high heat until shimmering but not smoking. Brown roast thoroughly on all sides, about 8 to 10 minutes. Transfer roast to plate; set aside.
Reduce heat to medium; add onion, carrot, and celery to pot and cook, stirring occasionally, until beginning to brown, 6 to 8 minutes. Add garlic and sugar; cook until fragrant, about 30 seconds. Add chicken and beef broths and thyme, scraping bottom of pan with wooden spoon to lessen browned bits.
Return roast and any accumulated juices to pot; add enough water to come halfway up side of roast. Bring liquid to simmer over medium heat, then place large piece of foil over pot and cover tightly with lid; transfer pot to oven.
Cook, turning roast every 30 minutes, until full tender and meat fork or sharp knife easily slips in and out of meat, 3-1/2 to 4 hours.
Transfer roast to carving board; tent with foil to keep warm. Allow liquid in pot to settle about 5 minutes, then use wide spoon to skim fat off surface; discard thyme sprig. Boil over high heat until reduced to about 1-1/2 cups, about 8 minutes. Add red wine and reduce again to 1-1/3 cups, about 2 minutes. Season to taste with salt and pepper.
Using carving knife, cut meat into 1/2-inch-thick slices, or pulls apart into large pieces; transfer meat to warmed serving platter and pour about 1/2 cup sauce over meat. Serve, passing remaining sauce separately.