In a large pan, heat 2 tablespoons of olive oil over medium heat.
Add the onion and sauté for about 3 minutes until translucent.
Add the garlic and cook for about 2 minutes, until fragrant.
Add the green pepper. Stir and sauté for about 5 minutes until tender.
Transfer the sautéed vegetables to a bowl and set aside.
Melt the vegan butter in the pan.
Add the mushrooms and stir to coat in the butter.
Add the paprika, ¼ teaspoon onion powder, ¼ teaspoon garlic powder, ½ teaspoon black pepper, and ½ teaspoon salt. Stir well. Sauté for about 7 minutes until the mushrooms are dark brown and reduce in size by half.
Add the green pepper, onion, and garlic mixture back to the pan and stir.
Add the soy sauce and Worcestershire sauce. Stir well. Continue to sauté for about 7 minutes until the liquid has thickened.
In the meantime, place the carrot, vegetable broth, remaining 3 tablespoons of olive oil, the lemon juice, nutritional yeast, remaining teaspoon of onion powder, remaining teaspoon of garlic powder, the tomato paste, remaining ½ teaspoon of salt, and remaining ½ teaspoon of pepper in a food processor. Blend until it forms a smooth and homogenous sauce.
Add the potato and blend until homogenous.
Fill the hoagies with the mushroom mixture.
Top the mushroom mixture with the vegan “cheese”, and parsley. Serve immediately.