At major fat seam, pull apart beef chuck-eye roast to create 2 smaller roasts, and cut away all large knobs of fat from each piece of meat (deboning knife works great). Season meat with Kosher salt, place on wire rack set in rimmed baking sheet, and let stand at room temperature for 1 hour. Adjust oven rack to lower-middle position and heat oven to 300 degrees. Melt butter in Dutch oven over medium heat. Add onions, stirring occasionally, until softened and beginning to brown, 8-10 minutes. Add carrot and celery. Continue to cook, stirring occasionally, for 5 minutes. Add garlic cloves and cook until fragrant, about 30 seconds. Stir in 1 cup beef broth, 1/2 cup dry red wine, 1 tablespoon tomato paste, bay leaf, and 1 sprig fresh thyme. Bring to simmer. Pat beef dry with paper towels and season with pepper. Tie 3 pieces of kitchen twine around each piece of meat to form even shape. Nestle meat on top of vegetables. Cover pot tightly with large piece of aluminum foil and cover with lid. Transfer pot to oven. Cook beef until fully tender and fork slips easily in and out of meat, 3 1/2 to 4 hours, turning meat halfway through cooking. Transfer roasts to carving board and tent loosely with foil. Strain liquid through fine-mesh strainer into 4-cup liquid measuring cup. Discard bay leaf and thyme sprig. Transfer vegetables to blender. Let liquid sit for 5 minutes, then skim fat. Add 1-2 cups beef broth to bring liquid amount up to 3 cups. Add liquid to blender and blend until smooth, about 2 minutes. Transfer sauce to medium saucepan and bring to simmer over medium heat. Meanwhile, remove twine from roasts and slice against grain into 1/2-inch-thick slices. Transfer meat to serving platter. Stir 1/4 cup dry red wine, 1/4 teaspoon chopped fresh thyme, and 1 tablespoon balsamic vinegar into gravy and season with salt and pepper to taste. Spoon half of gravy over meat; pass remaining gravy separately. ROOT VEGETABLE VARIATION: Add prepped root vegetables to pot after cooking beef for 3 hours. Continue to cook until beef is fully tender, 30 minutes to 1 hour longer. Transfer large pieces of carrot, parsnip and potato to serving platter using slotted spoon, cover tightly with aluminum foil, and proceed with recipe at Step 12. TROUBLESHOOTING: BUYING WRONG ROAST - Can result in dry meat, chalky meat, bland taste. Ignore stickers from butcher, avoid lean roasts from the round for they will become very tough and very dry if cooked this way; a fatty, well-marbled roast is better for this recipe. The most acceptable alternative to boneless chuck-eye roast is the top blade roast, but cooking time will vary. NOT HALVING ROAST - Can result in fatty meat, greasy gravy and the meat not cooked in the posted time. Be sure to remove the excess fat along the main seam. Separating the meat into 2 smaller well-trimmed roasts cuts the cooking time by about 1 hour. NOT SALTING ROAST - Can result in bland meat. The salting draws the moisture out of the meat, forming a shallow brine that during cooking seeps into the meat to season it throughout rather than just the exterior. Salt also enhances the beefy flavor of this cut. ADDING ALL BROTH AT OUTSET - Can result in unbrowned meat and bland taste. This recipe reserves most of the beef broth for the gravy, and counts on the beef sitting above the liquid in the pot, allowing the meat to brown without preliminary searing. NOT DEFATTING BRAISING LIQUID - Can result in greasy gravy. Be sure to follow steps 13-15 to get the full taste of the roasted vegetables in the gravy with a minimum of fat. The 5 minutes of resting time for the strained liquid allows the fat to rise to the surface, where it can be easily removed with a wide, shallow spoon. NOT PUREEING GRAVY IN BLENDER - Can result in thin and/or bland gravy. Pureeing the roasted vegetables ekes out the last bit of their flavor.