Adjust oven rack to middle position and 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 oven over medium-high heat until shimmering but not smoking.
Brown roast thoroughly on all sides, reducing heat if fat begins to smoke, 8 to 10 minutes.
Transfer roast to large 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 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 with lid; transfer pot to oven.
Cook, turning roast every 30 minutes, until roast is almost tender (sharp knife should meet little resistance), 3 hours. Do not allow to boil. If this means keeping the lid slightly ajar, then do that.
Add carrots, red potatoes, and celery to Dutch oven, submerging them in liquid. Continue to cook until vegetables are almost tender, about 40 minutes.
Transfer roast to carving board; tent with foil to keep warm.
Using slotted spoon, transfer vegetables to warmed serving bowl or platter.
Pour juices into gravy separator.
Melt butter in the bottom of the pan and add flour to make a roux. Stir until nutty brown.
Add juices (minus fat) back into the pot. Add wine and salt and pepper to taste; simmer over medium high heat, stirring, until gravy reaches desired consistency.
Using chef’s or carving knife, cut meat into 1/2-inch-thick slices or pull apart into large pieces; transfer to bowl or platter with vegetables and pour about 1/2 cup sauce over meat and vegetables. Serve, passing remaining sauce separately.