Preheat oven to 350°F and position rack in the middle.
In a microwave-safe bowl, combine ½ cup broth and mushrooms. Cover and microwave until steaming, about 1 minute. Let sit for 5 minutes.
Drain through a coffee filter-lined strainer, pressing to extract liquid. Reserve liquid and finely chop mushrooms.
Heat olive oil in a Dutch oven over medium heat until shimmering.
Add onion and cook until softened, about 5 minutes.
Stir in garlic and cook until fragrant, about 1 minute.
Add tomato paste, anchovies, and five-spice powder. Cook until darkened and a fond forms, 3 to 4 minutes.
Add wine, increase heat, and simmer until nearly dry, 2 to 4 minutes.
Stir in chopped tomatoes with juice, remaining ⅔ cup broth, reserved mushroom liquid, and chopped mushrooms. Bring to a simmer.
Season beef with ½ teaspoon salt and pepper. Add to the pot, cover, and transfer to the oven. Cook for 1 hour.
Uncover and cook until beef is tender, 1 to 1¼ hours longer.
Transfer beef to a cutting board and let cool 5 minutes. Shred with forks, discarding fat and connective tissue.
Skim excess fat from sauce. Return shredded beef to the pot.
Season to taste and serve.