Destem and wipe clean the portobellos. Cut them into ½" strips and set aside. Whisk together the ingredients for the portobello marinade. Transfer the marinade to a large ziplock bag. Add the portobello strips and shake the bag so that the portobellos are fully submerged. Let marinate while preparing the rest of the sandwiches.
Preheat the oven to 400 F.
Make the vegan cheese sauce. In a small saucepan over medium heat, melt butter. Add flour, whisking constantly, until a roux has formed. Add milk, ¼ cup at a time. Continue to whisk constantly. Bring to a boil and cook for about 4-5 minutes until the mixture begins to thicken - it should coat the back of a wooden spoon. Remove from heat and add Mozzarella Cutting Board Shreds, salt, and pepper. Stir in until the cheese has melted. Set aside.
Heat one tablespoon of oil over medium-low heat in a large skillet. Add onions and sauté for about 5 minutes, until onions are fragrant and translucent. Add peppers and button mushrooms and sauté for an additional 5 minutes until peppers cooked all the way through and slightly charred. Remove from heat and season with salt and pepper. Wipe clean the skillet. Add the remaining oil and increase the heat to medium. Add the marinated portobellos along with the marinade and sauté, stirring occasionally, for 3-5 minutes until the portobellos are crispy brown and charred on the outside.
Assemble the sandwiches by placing ¼th of the portobellos at the bottom of each hoagie roll. Top the onions/bell pepper mixture and a few dollops of vegan cheese sauce. Place on a baking sheet and bake 5 minutes until the bread is crispy and the cheese sauce is warm and melty. Remove from heat and serve immediately.