Heat 1 tablespoon of olive oil in a large Dutch oven or heavy-bottomed pot over medium-high heat. Season the thinly sliced steak with 1/2 tsp salt and 1/4 tsp black pepper.

Add the steak to the hot pot in a single layer, working in batches if necessary to avoid overcrowding. Sear for 1-2 minutes per side until browned, but still slightly pink in the center. Remove the cooked steak from the pot and set aside.

Reduce heat to medium. Add another tablespoon of olive oil to the same pot. Add the sliced yellow onion and green bell pepper. Sauté for 8-10 minutes, stirring occasionally, until softened and lightly caramelized. Add the minced garlic and cook for another minute until fragrant.

Push the vegetables to one side of the pot. Add the butter to the empty side and let it melt. Once melted, sprinkle in the all-purpose flour and whisk constantly for 1-2 minutes to create a roux, cooking out the raw flour taste.

Gradually whisk in the beef broth, a cup at a time, ensuring no lumps form. Stir in the Worcestershire sauce, dried oregano, and dried thyme. Bring the soup to a gentle simmer, then reduce heat to low, cover, and let it cook for 10-15 minutes, allowing the flavors to meld and the soup to thicken slightly.

Uncover the pot. Add the softened cream cheese cubes and shredded provolone cheese to the soup. Stir continuously until both cheeses are completely melted and incorporated, creating a smooth, creamy base.

Stir in the whole milk or half-and-half. Return the cooked steak to the pot and stir to combine. Heat through for 2-3 minutes, but do not boil after adding the milk and cheese.

Taste and adjust seasoning if needed. Ladle the hot Philly Cheesesteak Soup into bowls. Garnish with additional shredded provolone cheese and a sprinkle of fresh chopped parsley, if desired. Serve immediately with toasted baguette slices or crusty bread for dipping.