Preheat oven to 325°F.
Season and sear beef. Portion the chuck roast into equal 3 parts. Season with 1 ½ teaspoons salt and ½ teaspoon black pepper. Heat oil in a large 4-quart Dutch oven braiser over medium-high heat. Once hot, sear on both sides until a golden crust forms, about 4-5 minutes per side. Remove and set aside.
Sauté aromatics: In the same pan, cook the onions and celery, deglazing any burning spots with a splash of broth. Then, add the minced garlic for another 1 minute. Sprinkle 2 tablespoons of flour and stir to coat.
Add the liquids: Add the beef broth, dissolving the flour and deglazing any remaining browned bits. Next, add the balsamic vinegar, Worcestershire sauce, honey, and Dijon mustard. Bring to a simmer and season with salt and pepper to taste.
Braise: Return the seared beef and add the rosemary and thyme sprigs. Bring to a simmer. Cover tightly with the lid, then transfer to the oven for 1 hour 45 minutes.
Add a flour slurry: After that, separately in a bowl, whisk 2 tablespoons of flour with 2-3 tablespoons of water to make a slurry. Remove the pan from the oven, and add this slurry to the braising liquid. Give it a stir to combine.
Add the vegetables. Add the carrots and mushrooms, gently pushing them down into the liquid. Cover again, and return to the oven for another 1 hour.
Prep the puff pastry: 10 minutes before it's ready, roll out the puff pastry and cut 2-inch squares (I usually use a pastry wheel). Set aside or place it back in the fridge to keep cold.
Increase oven temperature to 400°F. Then, remove the pan from the oven.
Adjust gravy consistency: If needed, remove some liquid (I ended up removing about 1/4 cup). Remove the herb sprigs too. Note: Sometimes the meat releases extra juices when cooking. You want enough braising liquid to keep the filling hearty, but not so much that it's soupy.
Shred the beef: At this point, the beef should be very tender. Using a fork, shred into smaller chunks. Stir in the chopped Italian parsley until evenly combined.
Top with puff pastry: Arrange the puff pastry squares on top in an overlapping pattern. Brush the puff pastry with egg wash (1 egg whisked with ½ tablespoon heavy cream). Sprinkle the top with flaky sea salt.
Finish baking at 400°F for 20-25 minutes, or until the pastry is puffed up and golden and the filling is bubbling. Let it cool down for about 20 minutes before serving to allow the filling to settle a bit.