In a colander, sprinkle salt over cabbage and toss to coat. Let sit until cabbage has wilted slightly but still has a bit of crispness, 15 to 20 minutes. Squeeze out any excess moisture and transfer to a medium bowl.

Add scallions, cilantro, lime juice, olive oil, and seeds. Toss until thoroughly coated and combined. Set aside until ready to use.

Make Ahead: Slaw can be made 5 days ahead. Transfer to an airtight container and chill.

Preheat oven to 400ºF. Remove and discard stems from mushrooms. Slice mushrooms into ¼" pieces and/or rip oyster mushrooms into similar-sized pieces. Transfer to a sheet tray and add 2 tablespoons olive oil, 2 tablespoons soy sauce, and ½ teaspoon black pepper. Toss to combine.

Spread mushroom mixture in a single layer. Roast until mushrooms have released their moisture and begin to brown and crisp, 30 to 35 minutes.

While mushrooms roast, in a large skillet over medium-high heat, heat remaining 1 tablespoon plus 2 teaspoons olive oil until shimmering. Cook onion and white and pale-green scallion parts, stirring occasionally, until onion is translucent and beginning to turn golden on the edges, 6 to 7 minutes. Add ginger and garlic and cook, stirring, until fragrant, about 1 minute. Add sambal oelek and toss to combine. Cook, stirring, until incorporated, about 1 minute more.

Reduce heat to medium-low, stir in beans, and cook, stirring occasionally, until beans have softened slightly, 4 to 5 minutes. Remove from heat and gently mash with a wooden spoon or potato masher, leaving some beans partially intact. Let cool slightly, about 10 minutes.

In a food processor, pulse cashews until they resemble bread crumbs. Add panko, two-thirds of bean mixture, and one-half of roasted mushrooms to food processor.

In a large bowl, whisk egg, miso, vinegar, sesame oil, MSG, if using, and remaining 1 tablespoon soy sauce to combine. Pour into food processor. Pulse 10 to 15 times until a homogeneous, meat-like mixture forms.

Transfer to a large bowl. Fold in dark-green scallion parts and remaining roasted mushrooms and bean mixture. Cover bowl with plastic wrap and chill until cold, at least 20 minutes or up to overnight.

Form bean mixture into 4 patties for large burgers (or 6 patties for smaller burgers) and place on a plate. Season both sides of each patty with salt and black pepper.

In a large skillet over medium heat, heat vegetable oil until shimmering. Working in batches and adding more oil if necessary, sear patties, turning once, until browned and heated through, 2 to 3 minutes per side.

On cut sides of buns, spread mayonnaise. Top bottom bun with a burger, a big scoop of slaw, and a few slices of tomato. Close burgers with top bun.