Prepare the Mushrooms - Cut off the stems, then use a spoon to scoop out the brown gills. Gently clean any dirt off the surface using a damp paper towel.
Score the Tops - Use the tip of a paring knife to cut a ½-inch crosshatch pattern on the tops of the mushroom caps.
Make the Marinade - In a small bowl, whisk together olive oil, vinegar, soy sauce, minced garlic, salt, and pepper.
Marinate the Mushrooms - Add the mushrooms to a large plastic bag or baking dish. Turn to coat in the marinade and allow to sit for 30 to 60 minutes.
Prepare the Grill - Preheat the grill over medium-high heat 350 to 400ºF (177 to 204ºC). Clean and carefully grease the cooking grates with tongs and a paper towel dipped in oil.
Grill the Mushrooms - Remove the mushrooms from the marinade and place them on the hot grill, cap-side down. Cover and cook until the mushrooms have released their liquid with charred marks, 4 to 6 minutes. Flip and cook covered for 3 to 5 minutes. If adding cheese, place it on after flipping, then cover and cook until melted.
To Serve - Place the grilled portobellos on toasted burger buns and add desired toppings.