Preheat your oven to 400°F. In a large Dutch oven or a heavy-bottomed pot, warm 2 tablespoons of olive oil over medium-high heat. Add the chopped onion, diced carrot, celery seeds, and garlic. Cook for about 3 minutes, stirring occasionally, until the onion becomes soft and fragrant.

Sprinkle in 1/3 cup of flour and stir well to coat the vegetables. Add the Worcestershire sauce and a few dashes of hot sauce, continuing to stir for another 2 minutes to cook off the raw flour taste. Pour in the chicken broth and an additional 2 cups of water. Bring everything to a rolling boil, then reduce the heat to medium-low and let it simmer for about 20 minutes, or until the vegetables are fully tender.

While the soup simmers, prepare the croutons. Toss the cubed hoagie rolls with 1 tablespoon of olive oil, season with salt and pepper, and spread them out on a baking sheet. Bake in the preheated oven for about 7 minutes, or until they turn golden and crisp. Remove from the oven and set aside.

Once the vegetables have softened, blend the soup until smooth using a blender in batches, or an immersion blender directly in the pot. Return the puréed soup to the pot and bring it to a boil. Lower the heat to a gentle simmer for 5 more minutes.

In a bowl, combine the shredded cheddar and provolone cheeses with the remaining 2 tablespoons of flour. Gradually whisk this cheese mixture into the hot soup, stirring continuously until fully melted and integrated. For an extra silky texture, purée the soup once more and return it to the pot. Bring to a low boil, then take it off the heat.

In a separate large skillet, heat the remaining 2 tablespoons of olive oil over medium-high heat. Add the shaved beef and cook, stirring occasionally, until browned and heated through—about 3 minutes. Stir in the sliced pepperoncini and chopped parsley, and season to taste with salt and black pepper.

To serve, ladle the velvety soup into bowls. Top each serving with a generous spoonful of the seasoned beef, a handful of crispy croutons, and a sprinkle of fresh parsley.