Preheat the oven to 450 degrees F.
Pierce the roast all over with the tip of a paring knife and stuff one clove of garlic into each hole, followed by a few drops of hot sauce. Sprinkle on all sides with Essence and garlic powder and rub into the meat. Heat the oil in a Dutch oven over medium-high heat and brown the roast evenly on all sides, about 10 to 15 minutes. Add the chopped onion, celery, bell pepper, and enough water to come 1-inch up the sides of the pan. Add the teriyaki sauce, Worcestershire sauce, and stir to combine and loosen any browned bits on the bottom of the pan. Cover and bake for 45 minutes.
Reduce the oven temperature to 350 degrees F and add a more water as necessary to come 1-inch up the sides of the pan. Continue to cook until very tender, about 3 1/2 hours total cooking time.
Meanwhile, to make the gravy, melt 4 tablespoons of the butter in a medium saucepan. Whisk in the flour and cook, stirring constantly, over medium heat, until a milk chocolate-colored roux is formed, about 15 minutes. Add the onion, bell pepper, celery, garlic, and the white part of the green onions, and cook until the vegetables are soft but not browned, about 5 minutes. Whisk in the beef stock and heavy cream and cook until thickened, 10 to 15 minutes.
While the sauce is simmering, melt the remaining 2 tablespoons of butter in a medium skillet over medium-high heat. Add the mushrooms and cook, stirring, until tender and golden brown around the edges, about 5 minutes. Add the mushrooms and teriyaki sauce to the gravy and simmer until flavors come together, about 3 minutes. Remove from the heat and cover to keep warm.
Remove the roast from the oven and transfer to a platter. Cover loosely with foil to keep warm. Strain the pan juices through a fine mesh strainer into the gravy and simmer until heated through and thickened, about 5 minutes. Season with salt and pepper, to taste, and serve over the carved roast.