Prep and season the roast. Preheat the oven to 325°F (163ºC). Pat the chuck roast (1`) very dry with paper towels. Season all over with 4 teaspoons kosher salt, 2 teaspoons black pepper, and the garlic powder (2 teaspoons). Lightly dredge the roast in the 1/2 cup of flour, shaking off any excess.
Sear the beef. Heat 2 tablespoons olive oil in a large Dutch oven over medium-high heat until shimmering. Add the roast and sear on all sides until deeply browned, 3–4 minutes per side. Transfer the roast to a plate and set aside.
Build the flavor base. Reduce the heat to medium. Add the butter (2 tablespoons) to the pot (and the remaining 1 tablespoon oil if the pot looks dry). Once melted, add the chopped onion (1), chopped carrot (1), and chopped celery (1-2 ribs). Cook, stirring occasionally, until softened and lightly golden, about 5–7 minutes. Stir in the minced garlic (2 teaspoons) and tomato paste (1 tablespoon) and cook for 1 minute, stirring constantly, until fragrant and the tomato paste darkens slightly.
Deglaze and add liquids. Pour in the red wine (1 cup) and stir, scraping up any browned bits from the bottom of the pot. Let the wine simmer for 2–3 minutes. Whisk the beef base (1 tablespoon) into the beef stock (2 cups) and water (1/2 cup), then pour into the pot along with the Worcestershire sauce (2 teaspoons). Nestle in the halved head of garlic (cut side down), rosemary (2 sprigs), thyme (3 sprigs), and bay leaf (1).
Return the roast and start the braise. Return the seared roast and any accumulated juices to the pot. The liquid should come about halfway up the sides of the roast; if needed, add a splash more stock or water. Bring to a gentle simmer on the stovetop, then cover tightly and transfer to the oven. Braise for 2 hours.
Add the showpiece vegetables. After 2 hours, carefully remove the pot from the oven. Tuck the onion wedges, carrot chunks (3), potatoes (1 pound), and mushroom halves (1 pound) around the roast, spooning some of the braising liquid over the vegetables. Cover again and return to the oven. Continue braising for 60–75 minutes, or until the roast is very tender and shreds easily with a fork (an internal temperature of about 195–205°F) and the vegetables are just tender but still holding their shape.
Remove the roast + veggies. Transfer the roast and showpiece vegetables to a warm platter and tent loosely with foil. Fish out and discard the rosemary + thyme stems and bay leaf. Pull out the garlic head—squeeze the soft cloves into the pot and discard the peel.Blend the gravy base. With the pot still on the stove, use an immersion (stick) blender to puree the softened onion/carrot/celery right into the braising liquid until smooth and creamy. This turns all that cooked-down flavor into natural body and thickness.Simmer to thicken. Bring the blended gravy to a gentle simmer over medium heat and let it bubble uncovered for 5–10 minutes, stirring occasionally, until it reduces slightly and thickens to a gravy that coats the back of a spoon.Adjust consistency (easy fixes). - If it’s too thick, whisk in a splash of beef stock or water until it pours the way you like.- If it’s too thin, keep simmering uncovered a few minutes longer (reduction is your thickener here).Taste + finish. Taste and adjust with the remaining salt/pepper as needed, and add an extra dash of Worcestershire if you want more savory punch.Serve. Slice or shred the roast, arrange with the vegetables, and spoon that blended gravy over everything (serve extra at the table). Skim the fat. Let the braising liquid sit for a minute, then skim off excess fat from the surface with a spoon (this keeps the gravy from tasting greasy and helps it emulsify).