Step 1: Heat olive oil in a large skillet or cast-iron pan over medium heat. Add the thinly sliced onion and cook, stirring occasionally, for 8-10 minutes, or until softened and caramelized to a golden brown. Remove the onions from the pan and set aside.
Step 2: Increase the heat to medium-high. Add the shaved steak to the hot skillet, breaking it apart with a spoon. Season with salt, black pepper, and garlic powder. Cook for 3-5 minutes, stirring frequently, until the steak is browned and cooked through. Drain any excess liquid.
Step 3: Return the caramelized onions to the skillet with the cooked steak. Pour in the beef broth and stir to combine, scraping up any browned bits from the bottom of the pan. Cook for another 1-2 minutes until most of the broth has evaporated. Remove the skillet from the heat.
Step 4: Lay an egg roll wrapper on a clean, dry surface with one corner pointing towards you (like a diamond). Place about 2 tablespoons of the steak and onion mixture just below the center of the wrapper. Top the mixture with 1-2 slices or a small handful of provolone cheese. Do not overfill!
Step 5: Fold the bottom corner of the wrapper tightly over the filling. Fold in the left and right corners towards the center, creating an envelope shape. Brush the top corner of the wrapper with the beaten egg wash, then roll the egg roll tightly upwards to seal. Repeat with the remaining wrappers and filling.
Step 6: In a large, heavy-bottomed pot or Dutch oven, heat the vegetable oil over medium-high heat until it reaches 350°F (175°C). Carefully place 3-4 egg rolls into the hot oil, ensuring not to overcrowd the pan. Fry for 3-4 minutes, turning occasionally, until they are golden brown and crispy on all sides.
Step 7: Remove the crispy cheesesteak egg rolls with a slotted spoon and transfer them to a plate lined with paper towels to drain excess oil. Let them cool for a few minutes before serving. Serve hot with your favorite dipping sauce, such as ketchup, Sriracha mayo, or a warm cheese sauce.