Heat a large, nonstick skillet over medium-high heat. Add the oil and sauté the onion and garlic until aromatic, about 2 minutes. Stir in the sliced mushrooms and increase the heat to high. Add the soy sauce and salt, cooking until the mushrooms are tender and have released their liquid, about 3 minutes. Remove from the heat and let cool for 5 minutes.

Transfer the cooled mushroom mixture to a food processor along with the rice, miso paste, and pepper. Pulse until the mixture is minced and well incorporated but still retains some texture—avoid processing it into a smooth paste. Move the mixture to a large bowl and stir in the bread crumbs, flour, and sesame seeds with a spatula until uniform.

Form the mixture into four 1-inch-thick patties. If the mixture is too wet to hold its shape, add an extra tablespoon or two of flour.

Wipe out the skillet and heat a fresh layer of neutral oil over medium-high heat. When the oil is shimmering, add the patties and cook until the bottoms are deeply golden brown and crisp, about 5 minutes. Flip carefully and cook the other side for another 5 minutes.

Whisk the vegan mayonnaise, sriracha, and sesame seeds together in a small bowl until smooth.

Assemble the burgers by spreading the sriracha mayo onto the buns and layering with lettuce, tomatoes, onions, and the warm patties. Serve immediately while the patties are still hot and crisp.