Adjust oven rack to middle position and heat oven to 300°F.
Thoroughly pat roast dry with paper towels; sprinkle generously with salt and pepper.
Heat oil in 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 roast to a large plate; set aside.
Reduce heat to medium; add onion, carrot, and celery to pot and cook, stirring occasionally, until beginning to brown, 6- 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 loosen browned bits.
Return roast and any accumulated juices to pot; add enough water to come halfway up sides of roast.
Bring liquid to simmer over medium heat, then cover tightly and transfer pot to oven.
Cook, turning roast every 30 minutes until fully tender and meat fork slips in and out of meat very easily (3 1/2- 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 1/2 cups, about 2 minutes.
Season to taste with salt and pepper.
Cut meat into 1/2 inch slices, or pull apart in pieces; transfer meat to warmed serving platter and pour about 1/2 cup sauce over meat.
Serve, passing remaining sauce.
Note: Can add small red potatoes and sliced carrots and/or parsnips about 45 minutes before roast is finished.