Prepare the mushrooms by cutting off the stems, then use a spoon to scoop out the brown gills. Use a damp paper towel to gently clean any dirt off the surface.
Use the tip of a paring knife to cut a ½-inch crosshatch pattern on the tops of the mushroom caps, about 1/16-inch deep scores.
In a small bowl whisk together olive oil, vinegar, soy sauce, minced garlic, salt, and pepper.
Add mushrooms to a large plastic bag or baking dish. Turn to coat in the marinade and allow to sit for 30 to 60 minutes.
Preheat grill over medium-high heat 350 to 400ºF (177 to 204ºC).
Clean the grill and then carefully grease the cooking grates with tongs and a paper towel dipped in oil.
Remove the mushroom from the marinade then place on the hot grill, cap-side down.
Cover and cook until 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, cover and cook until melted.
Place the grilled portobellos on toasted burger buns and add desired toppings.