In a large ziptop bag, combine the soy sauce, balsamic vinegar, oil, garlic, Italian seasoning, and black pepper. Add the mushroom caps and seal the bag, removing as much air as possible. Gently swish the marinade around the mushrooms, then place the bag in a shallow dish. Allow the mushrooms to sit at room temperature for at least 30 minutes while you prepare any desired toppings, or refrigerate for up to 1 hour. Turn the bag once or twice while the mushrooms sit so that they are coated evenly.: The marinade should smell tangy, salty, and garlicky, with the balsamic lending a slight sweetness that balances the soy sauce. Letting the portobello mushroom caps sit at room temperature for thirty minutes allows them to draw in the liquid and flavors so the interior becomes seasoned, not just the surface. You will notice the mushrooms become slightly darker and supple as they hydrate. A common mistake is skimping on contact time and then overcooking to chase flavor which leads to a dry texture, so resist the urge to rush. Also, be cautious if refrigerating, because prolonged cold marination can firm the mushroom surface and slow the absorption rate. While the mushrooms marinate, prep your sliced tomato , arugula , and pesto , so assembly is seamless once everything is grilled. If the bag feels too full, use a shallow dish to keep them submerged and evenly coated; you want all surfaces to touch the marinade at least once or twice while they rest.
Heat an outdoor grill or indoor grill pan to medium heat. Once hot, brush the grate lightly with oil to prevent sticking, or lightly spray the grill pan with nonstick spray.: When the grill is at medium heat you should feel steady radiant warmth, and metal grates will take on a faint shimmer. Brushing the grate with a little oil creates a slick barrier that helps the mushrooms release easily and develop attractive grill marks. If using an indoor grill pan the sizzling sound will be slightly higher pitched and quicker to start; that means the surface is ready. Avoid setting the heat too high because mushrooms contain moisture and will release juices rapidly; intense heat can cause flare ups or uneven charring. A helpful tip is to oil the mushrooms lightly rather than over-oiling the grate, since excess oil can drip and cause smoke. Wipe the grates with a paper towel after oiling to remove globs that could char the mushrooms unevenly.
Remove the mushrooms from the marinade, shaking off any excess. Reserve the marinade in the bag for basting. Grill the mushrooms on both sides until tender, about 10 minutes total, brushing them with the marinade several times throughout.: As the mushrooms hit the hot grate you should hear a gentle, steady sizzle and smell an immediate savory perfume from the garlic and balsamic. Shaking off excess marinade prevents flare ups and concentrated sogginess while still leaving enough flavor on the surface to caramelize. Grill until the flesh yields when pressed with tongs, approximately ten minutes total, flipping once to produce even color. While grilling, baste with the reserved marinade to build layers of glaze and deepen the color; you will see the surface go from matte to glossy. One common error is overbasting early on which can add too much moisture and prevent browning, so wait until the first couple minutes of contact before the initial baste. If the mushrooms start releasing too much juice and steaming instead of searing, move them to a slightly cooler zone to encourage evaporation and more concentrated flavor.
When the mushrooms are in their final minute of cooking, turn two of them so that their undersides (the ones that previously had the gills and stem) are facing up. Top each with 1 slice of cheese, then stack a second mushroom on top, gill-side down, so that the cheese is in the middle. Place a second slice of cheese on top. Reduce the heat to low and cook just until the cheese is melted.: The final minute is when you create the melty, sandwich worthy center. Turning two mushrooms so their undersides face up creates a pocket for the cheese to sit directly against hot, juices filled surfaces, and stacking them traps heat so the cheese melts from both sides. When you place the cheese the sound softens and the aroma becomes creamier and richer. Lowering the heat prevents scorching the mushroom while allowing the cheese to melt gently; you want gooey, not burnt. A pitfall here is leaving the heat too high which melts the cheese unevenly and chars the mushroom stacks. For even melting, tent a lid or cover briefly if your grill allows, because that traps warm air and speeds melting without increasing direct heat.
During the last few minutes of cooking, if desired, grill sliced onions and toast the buns. To assemble, spread a thin layer of pesto on the cut sides of each bun. Place the arugula on the bottom bun, then top with a grilled portobello mushroom stack, onion, and tomato. Add the top bun. Enjoy immediately.: Grilling the sliced red onions softens their sharp edge and brings out a caramelized sweetness that pairs perfectly with the umami mushrooms. Toasting the whole wheat or brioche hamburger buns adds crunch and helps them stand up to the juices from the mushroom stack. When assembling, spreading a thin layer of pesto gives each bite an herbal lift and prevents the bun from becoming soggy by providing a barrier. The arugula offers peppery brightness under the mushroom, while the sliced tomato brings juiciness and acidity for balance. A common mistake is piling toppings too high or leaving the bun untoasted, which can result in a limp sandwich that falls apart. Press gently to compact the stack before serving and eat right away for the best contrast of textures and the melty cheese at its peak.