Heat 3 tablespoons butter and 1 tablespoon olive oil in a large skillet or Dutch oven over medium-low heat; add 3 sliced onions and a pinch of salt and cook, stirring occasionally, until deep golden brown, 35–45 minutes; reserve 1 cup of the finished onions for topping.

Pat the 3 lb roast very dry and season with 1 teaspoon kosher salt and 1/2 teaspoon black pepper; heat 1 tablespoon neutral oil in a Dutch oven over medium-high heat and sear the roast 3–4 minutes per side until a dark crust forms, including edges, then remove to a plate.

Add 1 tablespoon tomato paste to the Dutch oven and cook 1 minute, then stir in 1 cup beef stock, 1/2 cup water, 2 teaspoons Worcestershire sauce, and 1 tablespoon balsamic vinegar, scraping up browned bits; nestle the roast into the liquid and pile the remaining caramelized onions around and on top, adding extra stock so liquid reaches halfway up the roast sides; cover and transfer to a 300°F oven.

Braise until very tender and an instant-read thermometer in the center reads 200–205°F and a fork slides in with little resistance, about 2 1/2–3 hours depending on roast shape; begin checking at 2 hours.

Remove the roast and rest under foil 15 minutes; skim fat from liquid and reduce over medium-high heat until slightly thickened, 5–8 minutes; shred or pull the roast into large chunks and return to the sauce, spooning reserved onions over the top; for a browned finish, broil 2–3 minutes, watching closely.

Serve warm with mashed potatoes, noodles, or roasted vegetables.